TripleDNN

Current Directory: /usr/share/perl5/Math

File Name	Size	Actions
📁 BigFloat	Folder	Edit \| Delete \| Rename \| Download
📁 BigInt	Folder	Edit \| Delete \| Rename \| Download
BigFloat.pm	133802 bytes	Edit \| Delete \| Rename \| Download
BigInt.pm	158511 bytes	Edit \| Delete \| Rename \| Download
BigRat.pm	52767 bytes	Edit \| Delete \| Rename \| Download
Complex.pm	49439 bytes	Edit \| Delete \| Rename \| Download
Trig.pm	21374 bytes	Edit \| Delete \| Rename \| Download

#
# "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;

# anything older is untested, and unlikely to work
use 5.006;
use strict;
use Carp ();

use Math::BigFloat;
use vars qw($VERSION @ISA $upgrade $downgrade
            $accuracy $precision $round_mode $div_scale $_trap_nan $_trap_inf);

@ISA = qw(Math::BigFloat);

$VERSION = '0.2603';
$VERSION = eval $VERSION;

# inherit overload from Math::BigFloat, but disable the bitwise ops that don't
# make much sense for rationals unless they're truncated or something first

use overload
    map {
	my $op = $_;
	($op => sub {
	    Carp::croak("bitwise operation $op not supported in Math::BigRat");
	});
    } qw(& | ^ ~ << >> &= |= ^= <<= >>=);

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 MBF has a different $MBI variable than we, because MBF 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_int = \&as_number;
  *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 package we are using for our private parts, defaults to:
# Math::BigInt->config()->{lib}
my $MBI = '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_from_float
  {
  # turn a single float input into a rational number (like '0.1')
  my ($self,$f) = @_;

return $self->bnan() if $f->is_nan();
  return $self->binf($f->{sign}) if $f->{sign} =~ /^[+-]inf$/;

$self->{_n} = $MBI->_copy( $f->{_m} );	# mantissa
  $self->{_d} = $MBI->_one();
  $self->{sign} = $f->{sign} || '+';
  if ($f->{_es} eq '-')
    {
    # something like Math::BigRat->new('0.1');
    # 1 / 1 => 1/10
    $MBI->_lsft ( $self->{_d}, $f->{_e} ,10);
    }
  else
    {
    # something like Math::BigRat->new('10');
    # 1 / 1 => 10/1
    $MBI->_lsft ( $self->{_n}, $f->{_e} ,10) unless
      $MBI->_is_zero($f->{_e});
    }
  $self;
  }

sub new
  {
  # create a Math::BigRat
  my $class = shift;

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

my $self = { }; bless $self,$class;

# input like (BigInt) or (BigFloat):
  if ((!defined $d) && (ref $n) && (!$n->isa('Math::BigRat')))
    {
    if ($n->isa('Math::BigFloat'))
      {
      $self->_new_from_float($n);
      }
    if ($n->isa('Math::BigInt'))
      {
      # TODO: trap NaN, inf
      $self->{_n} = $MBI->_copy($n->{value});		# "mantissa" = N
      $self->{_d} = $MBI->_one();			# d => 1
      $self->{sign} = $n->{sign};
      }
    if ($n->isa('Math::BigInt::Lite'))
      {
      # TODO: trap NaN, inf
      $self->{sign} = '+'; $self->{sign} = '-' if $$n < 0;
      $self->{_n} = $MBI->_new(abs($$n));		# "mantissa" = N
      $self->{_d} = $MBI->_one();			# d => 1
      }
    return $self->bnorm();				# normalize (120/1 => 12/10)
    }

# input like (BigInt,BigInt) or (BigLite,BigLite):
  if (ref($d) && ref($n))
    {
    # do N first (for $self->{sign}):
    if ($n->isa('Math::BigInt'))
      {
      # TODO: trap NaN, inf
      $self->{_n} = $MBI->_copy($n->{value});		# "mantissa" = N
      $self->{sign} = $n->{sign};
      }
    elsif ($n->isa('Math::BigInt::Lite'))
      {
      # TODO: trap NaN, inf
      $self->{sign} = '+'; $self->{sign} = '-' if $$n < 0;
      $self->{_n} = $MBI->_new(abs($$n));		# "mantissa" = $n
      }
    else
      {
      require Carp;
      Carp::croak(ref($n) . " is not a recognized object format for Math::BigRat->new");
      }
    # now D:
    if ($d->isa('Math::BigInt'))
      {
      # TODO: trap NaN, inf
      $self->{_d} = $MBI->_copy($d->{value});		# "mantissa" = D
      # +/+ or -/- => +, +/- or -/+ => -
      $self->{sign} = $d->{sign} ne $self->{sign} ? '-' : '+';
      }
    elsif ($d->isa('Math::BigInt::Lite'))
      {
      # TODO: trap NaN, inf
      $self->{_d} = $MBI->_new(abs($$d));		# "mantissa" = D
      my $ds = '+'; $ds = '-' if $$d < 0;
      # +/+ or -/- => +, +/- or -/+ => -
      $self->{sign} = $ds ne $self->{sign} ? '-' : '+';
      }
    else
      {
      require Carp;
      Carp::croak(ref($d) . " is not a recognized object format for Math::BigRat->new");
      }
    return $self->bnorm();				# normalize (120/1 => 12/10)
    }
  return $n->copy() if ref $n;				# already a BigRat

if (!defined $n)
    {
    $self->{_n} = $MBI->_zero();			# undef => 0
    $self->{_d} = $MBI->_one();
    $self->{sign} = '+';
    return $self;
    }

# string input with / delimiter
  if ($n =~ /\s*\/\s*/)
    {
    return $class->bnan() if $n =~ /\/.*\//;	# 1/2/3 isn't valid
    return $class->bnan() if $n =~ /\/\s*$/;	# 1/ isn't valid
    ($n,$d) = split (/\//,$n);
    # try as BigFloats first
    if (($n =~ /[\.eE]/) || ($d =~ /[\.eE]/))
      {
      local $Math::BigFloat::accuracy = undef;
      local $Math::BigFloat::precision = undef;

# one of them looks like a float
      my $nf = Math::BigFloat->new($n,undef,undef);
      $self->{sign} = '+';
      return $self->bnan() if $nf->is_nan();

$self->{_n} = $MBI->_copy( $nf->{_m} );	# get mantissa

# now correct $self->{_n} due to $n
      my $f = Math::BigFloat->new($d,undef,undef);
      return $self->bnan() if $f->is_nan();
      $self->{_d} = $MBI->_copy( $f->{_m} );

# calculate the difference between nE and dE
      my $diff_e = $nf->exponent()->bsub( $f->exponent);
      if ($diff_e->is_negative())
	{
        # < 0: mul d with it
        $MBI->_lsft( $self->{_d}, $MBI->_new( $diff_e->babs()), 10);
	}
      elsif (!$diff_e->is_zero())
        {
        # > 0: mul n with it
        $MBI->_lsft( $self->{_n}, $MBI->_new( $diff_e), 10);
        }
      }
    else
      {
      # both d and n look like (big)ints

$self->{sign} = '+';					# no sign => '+'
      $self->{_n} = undef;
      $self->{_d} = undef;
      if ($n =~ /^([+-]?)0*([0-9]+)\z/)				# first part ok?
	{
	$self->{sign} = $1 || '+';				# no sign => '+'
	$self->{_n} = $MBI->_new($2 || 0);
        }

if ($d =~ /^([+-]?)0*([0-9]+)\z/)				# second part ok?
	{
	$self->{sign} =~ tr/+-/-+/ if ($1 || '') eq '-';	# negate if second part neg.
	$self->{_d} = $MBI->_new($2 || 0);
        }

if (!defined $self->{_n} || !defined $self->{_d})
	{
        $d = Math::BigInt->new($d,undef,undef) unless ref $d;
        $n = Math::BigInt->new($n,undef,undef) unless ref $n;

if ($n->{sign} =~ /^[+-]$/ && $d->{sign} =~ /^[+-]$/)
	  {
	  # both parts are ok as integers (wierd things like ' 1e0'
          $self->{_n} = $MBI->_copy($n->{value});
          $self->{_d} = $MBI->_copy($d->{value});
          $self->{sign} = $n->{sign};
          $self->{sign} =~ tr/+-/-+/ if $d->{sign} eq '-';	# -1/-2 => 1/2
          return $self->bnorm();
	  }

$self->{sign} = '+';					# a default sign
        return $self->bnan() if $n->is_nan() || $d->is_nan();

# handle inf cases:
        if ($n->is_inf() || $d->is_inf())
	  {
	  if ($n->is_inf())
	    {
	    return $self->bnan() if $d->is_inf();		# both are inf => NaN
	    my $s = '+'; 		# '+inf/+123' or '-inf/-123'
	    $s = '-' if substr($n->{sign},0,1) ne $d->{sign};
	    # +-inf/123 => +-inf
	    return $self->binf($s);
	    }
          # 123/inf => 0
          return $self->bzero();
	  }
	}
      }

return $self->bnorm();
    }

# simple string input
  if (($n =~ /[\.eE]/) && $n !~ /^0x/)
    {
    # looks like a float, quacks like a float, so probably is a float
    $self->{sign} = 'NaN';
    local $Math::BigFloat::accuracy = undef;
    local $Math::BigFloat::precision = undef;
    $self->_new_from_float(Math::BigFloat->new($n,undef,undef));
    }
  else
    {
    # for simple forms, use $MBI directly
    if ($n =~ /^([+-]?)0*([0-9]+)\z/)
      {
      $self->{sign} = $1 || '+';
      $self->{_n} = $MBI->_new($2 || 0);
      $self->{_d} = $MBI->_one();
      }
    else
      {
      my $n = Math::BigInt->new($n,undef,undef);
      $self->{_n} = $MBI->_copy($n->{value});
      $self->{_d} = $MBI->_one();
      $self->{sign} = $n->{sign};
      return $self->bnan() if $self->{sign} eq 'NaN';
      return $self->binf($self->{sign}) if $self->{sign} =~ /^[+-]inf$/;
      }
    }
  $self->bnorm();
  }

sub copy
  {
  # if two arguments, the first one is the class to "swallow" subclasses
  my ($c,$x) = @_;

if (scalar @_ == 1)
    {
    $x = $_[0];
    $c = ref($x);
    }
  return unless ref($x); # only for objects

my $self = bless {}, $c;

$self->{sign} = $x->{sign};
  $self->{_d} = $MBI->_copy($x->{_d});
  $self->{_n} = $MBI->_copy($x->{_n});
  $self->{_a} = $x->{_a} if defined $x->{_a};
  $self->{_p} = $x->{_p} if defined $x->{_p};
  $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} = $MBI;
  $cfg;
  }

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

sub bstr
  {
  my ($self,$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 . $MBI->_str($x->{_n}) if $MBI->_is_one($x->{_d});
  $s . $MBI->_str($x->{_n}) . '/' . $MBI->_str($x->{_d});
  }

sub bsstr
  {
  my ($self,$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 . $MBI->_str($x->{_n}) . '/' . $MBI->_str($x->{_d});
  }

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

# Both parts must be objects of whatever we are using today.
  if ( my $c = $MBI->_check($x->{_n}) )
    {
    require Carp; Carp::croak ("n did not pass the self-check ($c) in bnorm()");
    }
  if ( my $c = $MBI->_check($x->{_d}) )
    {
    require Carp; Carp::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 ($MBI->_is_zero($x->{_n}))
    {
    $x->{sign} = '+';					# never leave a -0
    $x->{_d} = $MBI->_one() unless $MBI->_is_one($x->{_d});
    return $x;
    }

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

# reduce other numbers
  my $gcd = $MBI->_copy($x->{_n});
  $gcd = $MBI->_gcd($gcd,$x->{_d});

if (!$MBI->_is_one($gcd))
    {
    $x->{_n} = $MBI->_div($x->{_n},$gcd);
    $x->{_d} = $MBI->_div($x->{_d},$gcd);
    }
  $x;
  }

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

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

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

# for +0 dont negate (to have always normalized +0). Does nothing for 'NaN'
  $x->{sign} =~ tr/+-/-+/ unless ($x->{sign} eq '+' && $MBI->_is_zero($x->{_n}));
  $x;
  }

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

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

if ($_trap_nan)
    {
    require Carp;
    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} = $MBI->_zero() unless defined $self->{_d};
    $self->{_n} = $MBI->_zero() unless defined $self->{_n};
    Carp::croak ("Tried to set $self to NaN in $class\::_bnan()");
    }
  $self->{_n} = $MBI->_zero();
  $self->{_d} = $MBI->_zero();
  }

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

if ($_trap_inf)
    {
    require Carp;
    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} = $MBI->_zero() unless defined $self->{_d};
    $self->{_n} = $MBI->_zero() unless defined $self->{_n};
    Carp::croak ("Tried to set $self to inf in $class\::_binf()");
    }
  $self->{_n} = $MBI->_zero();
  $self->{_d} = $MBI->_zero();
  }

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

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

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

sub badd
  {
  # add two rational numbers

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

# +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} = $MBI->_mul( $x->{_n}, $y->{_d});

# 7 * 4
  my $m = $MBI->_mul( $MBI->_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} = $MBI->_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 '+' && $MBI->_is_zero($x->{_n});	# not -0
  $x->badd($y,@r);				# does norm and round
  $x->{sign} =~ tr/+-/-+/
    unless $x->{sign} eq '+' && $MBI->_is_zero($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 ? ($x,$self->bzero()) : $x if $x->is_zero();

# XXX TODO:
  # 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.

#  1   2    1 * 2    2    1
  #  - * - =  -----  = -  = -
  #  4   3    4 * 3    12   6

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

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

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

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

return $self->_div_inf($x,$y)
   if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/) || $y->is_zero());

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

# 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.

# 1     1    1   3
  # -  /  - == - * -
  # 4     3    4   1

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

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

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

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

return $self->_div_inf($x,$y)
   if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/) || $y->is_zero());

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

# compute $x - $y * floor($x/$y), keeping the sign of $x

# copy x to u, make it positive and then do a normal division ($u/$y)
  my $u = bless { sign => '+' }, $self;
  $u->{_n} = $MBI->_mul( $MBI->_copy($x->{_n}), $y->{_d} );
  $u->{_d} = $MBI->_mul( $MBI->_copy($x->{_d}), $y->{_n} );

# compute floor(u)
  if (! $MBI->_is_one($u->{_d}))
    {
    $u->{_n} = $MBI->_div($u->{_n},$u->{_d});	# 22/7 => 3/1 w/ truncate
    # no need to set $u->{_d} to 1, since below we set it to $y->{_d} anyway
    }

# now compute $y * $u
  $u->{_d} = $MBI->_copy($y->{_d});		# 1 * $y->{_d}, see floor above
  $u->{_n} = $MBI->_mul($u->{_n},$y->{_n});

my $xsign = $x->{sign}; $x->{sign} = '+';	# remember sign and make x positive
  # compute $x - $u
  $x->bsub($u);
  $x->{sign} = $xsign;				# put sign back

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

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

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

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

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

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

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

if ($x->{sign} eq '-')
    {
    if ($MBI->_acmp($x->{_n},$x->{_d}) < 0)
      {
      # -1/3 ++ => 2/3 (overflow at 0)
      $x->{_n} = $MBI->_sub( $MBI->_copy($x->{_d}), $x->{_n});
      $x->{sign} = '+';
      }
    else
      {
      $x->{_n} = $MBI->_sub($x->{_n}, $x->{_d}); 	# -5/2 => -3/2
      }
    }
  else
    {
    $x->{_n} = $MBI->_add($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 ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);

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

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

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

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

my $sign = $_[2] || ''; $sign = '+' if $sign ne '-';
  return 1
   if ($x->{sign} eq $sign && $MBI->_is_one($x->{_n}) && $MBI->_is_one($x->{_d}));
  0;
  }

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

return 1 if ($x->{sign} =~ /^[+-]$/) &&		# NaN & +-inf aren't
    ($MBI->_is_one($x->{_d}) && $MBI->_is_odd($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 ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);

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

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

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

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

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

sub denominator
  {
  my ($self,$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($MBI->_str($x->{_d}));
  }

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

return $x if $x->{sign} =~ /^[+-]inf$/;       # -inf/+inf ** x
  return $x->bnan() if $x->{sign} eq $nan || $y->{sign} eq $nan;
  return $x->bone(@r) if $y->is_zero();
  return $x->round(@r) if $x->is_one() || $y->is_one();

if ($x->{sign} eq '-' && $MBI->_is_one($x->{_n}) && $MBI->_is_one($x->{_d}))
    {
    # if $x == -1 and odd/even y => +1/-1
    return $y->is_odd() ? $x->round(@r) : $x->babs()->round(@r);
    # my Casio FX-5500L has a bug here: -1 ** 2 is -1, but -1 * -1 is 1;
    }
  # 1 ** -y => 1 / (1 ** |y|)
  # so do test for negative $y after above's clause

return $x->round(@r) if $x->is_zero();  # 0**y => 0 (if not y <= 0)

# shortcut if y == 1/N (is then sqrt() respective broot())
  if ($MBI->_is_one($y->{_n}))
    {
    return $x->bsqrt(@r) if $MBI->_is_two($y->{_d});	# 1/2 => sqrt
    return $x->broot($MBI->_str($y->{_d}),@r);		# 1/N => root(N)
    }

# shortcut y/1 (and/or x/1)
  if ($MBI->_is_one($y->{_d}))
    {
    # shortcut for x/1 and y/1
    if ($MBI->_is_one($x->{_d}))
      {
      $x->{_n} = $MBI->_pow($x->{_n},$y->{_n});		# x/1 ** y/1 => (x ** y)/1
      if ($y->{sign} eq '-')
        {
        # 0.2 ** -3 => 1/(0.2 ** 3)
        ($x->{_n},$x->{_d}) = ($x->{_d},$x->{_n});	# swap
        }
      # correct sign; + ** + => +
      if ($x->{sign} eq '-')
        {
        # - * - => +, - * - * - => -
        $x->{sign} = '+' if $MBI->_is_even($y->{_n});
        }
      return $x->round(@r);
      }
    # x/z ** y/1
    $x->{_n} = $MBI->_pow($x->{_n},$y->{_n});		# 5/2 ** y/1 => 5 ** y / 2 ** y
    $x->{_d} = $MBI->_pow($x->{_d},$y->{_n});
    if ($y->{sign} eq '-')
      {
      # 0.2 ** -3 => 1/(0.2 ** 3)
      ($x->{_n},$x->{_d}) = ($x->{_d},$x->{_n});	# swap
      }
    # correct sign; + ** + => +
    if ($x->{sign} eq '-')
      {
      # - * - => +, - * - * - => -
      $x->{sign} = '+' if $MBI->_is_even($y->{_n});
      }
    return $x->round(@r);
    }

#  print STDERR "# $x $y\n";

# otherwise:

#      n/d     n  ______________
  # a/b       =  -\/  (a/b) ** d

# (a/b) ** n == (a ** n) / (b ** n)
  $MBI->_pow($x->{_n}, $y->{_n} );
  $MBI->_pow($x->{_d}, $y->{_n} );

return $x->broot($MBI->_str($y->{_d}),@r);		# n/d => root(n)
  }

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

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

# blog(1,Y) => 0
  return $x->bzero() if $x->is_one() && $y->{sign} eq '+';

# $x <= 0 => NaN
  return $x->bnan() if $x->is_zero() || $x->{sign} ne '+' || $y->{sign} ne '+';

if ($x->is_int() && $y->is_int())
    {
    return $self->new($x->as_number()->blog($y->as_number(),@r));
    }

# do it with floats
  $x->_new_from_float( $x->_as_float()->blog(Math::BigFloat->new("$y"),@r) );
  }

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

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

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] = $self->div_scale();	# and round to it as accuracy
    $params[1] = undef;			# P = undef
    $scale = $params[0]+4;		# at least four more for proper round
    $params[2] = $r[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} = $MBI->_new("90933395208605785401971970164779391644753259799242");
    $x->{_d} = $MBI->_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 = $MBI->_new("90933395208605785401971970164779391644753259799242");
    my $F = $MBI->_new(42); my $step = 42;

# Compute how many steps we need to take to get $A and $B sufficiently big
    my $steps = Math::BigFloat::_len_to_steps($scale - 4);
#    print STDERR "# Doing $steps steps for ", $scale-4, " digits\n";
    while ($step++ <= $steps)
      {
      # calculate $a * $f + 1
      $A = $MBI->_mul($A, $F);
      $A = $MBI->_inc($A);
      # increment f
      $F = $MBI->_inc($F);
      }
    # compute $B as factorial of $steps (this is faster than doing it manually)
    my $B = $MBI->_fac($MBI->_new($steps));

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

$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 ($self,$x,$y,@r) = (ref($_[0]),@_);

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

# do it with floats
  $x->_new_from_float( $x->_as_float()->bnok(Math::BigFloat->new("$y"),@r) );
  }

sub _float_from_part
  {
  my $x = shift;

my $f = Math::BigFloat->bzero();
  $f->{_m} = $MBI->_copy($x);
  $f->{_e} = $MBI->_zero();

$f;
  }

sub _as_float
  {
  my $x = shift;

local $Math::BigFloat::upgrade = undef;
  local $Math::BigFloat::accuracy = undef;
  local $Math::BigFloat::precision = undef;
  # 22/7 => 3.142857143..

my $a = $x->accuracy() || 0;
  if ($a != 0 || !$MBI->_is_one($x->{_d}))
    {
    # n/d
    return scalar Math::BigFloat->new($x->{sign} . $MBI->_str($x->{_n}))->bdiv( $MBI->_str($x->{_d}), $x->accuracy());
    }
  # just n
  Math::BigFloat->new($x->{sign} . $MBI->_str($x->{_n}));
  }

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

if ($x->is_int() && $y->is_int())
    {
    return $self->new($x->as_number()->broot($y->as_number(),@r));
    }

# do it with floats
  $x->_new_from_float( $x->_as_float()->broot($y->_as_float(),@r) )->bnorm()->bround(@r);
  }

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

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

if ($x->is_int() && $y->is_int() && $m->is_int())
    {
    return $self->new($x->as_number()->bmodpow($y->as_number(),$m,@r));
    }

warn ("bmodpow() not fully implemented");
  $x->bnan();
  }

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

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

if ($x->is_int() && $y->is_int())
    {
    return $self->new($x->as_number()->bmodinv($y->as_number(),@r));
    }

warn ("bmodinv() not fully implemented");
  $x->bnan();
  }

sub bsqrt
  {
  my ($self,$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();

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;

$x->{_n} = _float_from_part( $x->{_n} )->bsqrt();
  $x->{_d} = _float_from_part( $x->{_d} )->bsqrt();

# XXX TODO: we probably can optimze this:

# if sqrt(D) was not integer
  if ($x->{_d}->{_es} ne '+')
    {
    $x->{_n}->blsft($x->{_d}->exponent()->babs(),10);	# 7.1/4.51 => 7.1/45.1
    $x->{_d} = $MBI->_copy( $x->{_d}->{_m} );		# 7.1/45.1 => 71/45.1
    }
  # if sqrt(N) was not integer
  if ($x->{_n}->{_es} ne '+')
    {
    $x->{_d}->blsft($x->{_n}->exponent()->babs(),10);	# 71/45.1 => 710/45.1
    $x->{_n} = $MBI->_copy( $x->{_n}->{_m} );		# 710/45.1 => 710/451
    }

# convert parts to $MBI again
  $x->{_n} = $MBI->_lsft( $MBI->_copy( $x->{_n}->{_m} ), $x->{_n}->{_e}, 10)
    if ref($x->{_n}) ne $MBI && ref($x->{_n}) ne 'ARRAY';
  $x->{_d} = $MBI->_lsft( $MBI->_copy( $x->{_d}->{_m} ), $x->{_d}->{_e}, 10)
    if ref($x->{_d}) ne $MBI && ref($x->{_d}) ne 'ARRAY';

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

sub blsft
  {
  my ($self,$x,$y,$b,@r) = objectify(3,@_);

$b = 2 unless defined $b;
  $b = $self->new($b) unless ref ($b);
  $x->bmul( $b->copy()->bpow($y), @r);
  $x;
  }

sub brsft
  {
  my ($self,$x,$y,$b,@r) = objectify(3,@_);

$b = 2 unless defined $b;
  $b = $self->new($b) unless ref ($b);
  $x->bdiv( $b->copy()->bpow($y), @r);
  $x;
  }

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

sub round
  {
  $_[0];
  }

sub bround
  {
  $_[0];
  }

sub bfround
  {
  $_[0];
  }

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

sub bcmp
  {
  # compare two signed numbers

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

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

# shortcut
  my $xz = $MBI->_is_zero($x->{_n});
  my $yz = $MBI->_is_zero($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 = $MBI->_mul( $MBI->_copy($x->{_n}), $y->{_d});
  my $u = $MBI->_mul( $MBI->_copy($y->{_n}), $x->{_d});

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

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

##############################################################################
# output conversation

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

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

# N/1 => N
  my $neg = ''; $neg = '-' if $x->{sign} eq '-';
  return $neg . $MBI->_num($x->{_n}) if $MBI->_is_one($x->{_d});

$x->_as_float()->numify() + 0.0;
  }

sub as_number
  {
  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} = $MBI->_div( $MBI->_copy($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 ($self,$x,@r) = (ref($_[0]),@_);
  # objectify is costly, so avoid it
  ($self,$x,@r) = objectify(1,$class,@_) unless ref $_[0];

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

my $u = Math::BigFloat->bzero();
  $u->{sign} = $x->{sign};
  # n
  $u->{_m} = $MBI->_copy($x->{_n});
  $u->{_e} = $MBI->_zero();
  $u->bdiv( $MBI->_str($x->{_d}), @r);
  # return $u
  $u;
  }

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

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

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

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

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

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

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

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

my $s = $x->{sign}; $s = '' if $s eq '+';
  $s . $MBI->_as_oct($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 $self = 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 { $self->new(shift); };
      }
#    elsif ($_[$i] eq 'upgrade')
#      {
#     # this causes upgrading
#      $upgrade = $_[$i+1];		# or undef to disable
#      $i++;
#      }
    elsif ($_[$i] eq 'downgrade')
      {
      # this causes downgrading
      $downgrade = $_[$i+1];		# or undef to disable
      $i++;
      }
    elsif ($_[$i] =~ /^(lib|try|only)\z/)
      {
      $lib = $_[$i+1] || '';		# default Calc
      $try = $1;			# lib, try or only
      $i++;
      }
    elsif ($_[$i] eq 'with')
      {
      # this argument is no longer used
      #$MBI = $_[$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 '';

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

$MBI = Math::BigFloat->config()->{lib};

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

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

__END__

=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.

=head2 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

=head2 numerator()

$n = $x->numerator();

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

=head2 denominator()

$d = $x->denominator();

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

=head2 parts()

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

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

=head2 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

=head2 as_int()/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()>.

=head2 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).

=head2 as_hex()

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

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

=head2 as_bin()

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

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

=head2 as_oct()

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

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

=head2 from_hex()/from_bin()/from_oct()

my $h = Math::BigRat->from_hex('0x10');
	my $b = Math::BigRat->from_bin('0b10000000');
	my $o = Math::BigRat->from_oct('020');

Create a BigRat from an hexadecimal, binary or octal number
in string form.

=head2 length()

$len = $x->length();

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

=head2 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.

=head2 bnorm()

$x->bnorm();

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

=head2 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.

=head2 bround()/round()/bfround()

Are not yet implemented.

=head2 bmod()

use Math::BigRat;
	my $x = Math::BigRat->new('7/4');
	my $y = Math::BigRat->new('4/3');
	print $x->bmod($y);

Set $x to the remainder of the division of $x by $y.

=head2 bneg()

$x->bneg();

Used to negate the object in-place.

=head2 is_one()

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

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

=head2 is_zero()

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

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

=head2 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()>.

=head2 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()>.

=head2 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.

=head2 is_odd()

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

Return true if $x is odd, otherwise false.

=head2 is_even()

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

Return true if $x is even, otherwise false.

=head2 bceil()

$x->bceil();

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

=head2 bfloor()

$x->bfloor();

Truncate $x to an integer value.

=head2 bsqrt()

$x->bsqrt();

Calculate the square root of $x.

=head2 broot()

$x->broot($n);

Calculate the N'th root of $x.

=head2 badd()/bmul()/bsub()/bdiv()/bdec()/binc()

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

=head2 copy()

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

Makes a deep copy of the object.

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

=head2 bstr()/bsstr()

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

Return a string representing this object.

=head2 bacmp()/bcmp()

Used to compare numbers.

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

=head2 blsft()/brsft()

Used to shift numbers left/right.

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

=head2 bpow()

$x->bpow($y);

Compute $x ** $y.

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

=head2 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).