    \begin{code}
    {-# OPTIONS_GHC -fno-implicit-prelude #-}
    -----------------------------------------------------------------------------
    -- |
    -- Module      :  GHC.Real
    -- Copyright   :  (c) The FFI Task Force, 1994-2002
    -- License     :  see libraries/base/LICENSE
    --
    -- Maintainer  :  cvs-ghc@haskell.org
    -- Stability   :  internal
    -- Portability :  non-portable (GHC Extensions)
    --
    -- The types 'Ratio' and 'Rational', and the classes 'Real', 'Fractional',
    -- 'Integral', and 'RealFrac'.
    --
    -----------------------------------------------------------------------------

    -- #hide
    module GHC.Real where

    import GHC.Base
    import GHC.Num
    import GHC.List
    import GHC.Enum
    import GHC.Show

    infixr 8  ^, ^^
    infixl 7  /, `quot`, `rem`, `div`, `mod`
    infixl 7  %

    default ()              -- Double isn't available yet,
                            -- and we shouldn't be using defaults anyway
    \end{code}

    %*********************************************************
    %*                                                      *
    \subsection{The @Ratio@ and @Rational@ types}
    %*                                                      *
    %*********************************************************

    \begin{code}
    -- | Rational numbers, with numerator and denominator of some 'Integral' type.
    data  (Integral a)      => Ratio a = !a :% !a  deriving (Eq)

    -- | Arbitrary-precision rational numbers, represented as a ratio of
    -- two 'Integer' values.  A rational number may be constructed using
    -- the '%' operator.
    type  Rational          =  Ratio Integer

    ratioPrec, ratioPrec1 :: Int
    ratioPrec  = 7  -- Precedence of ':%' constructor
    ratioPrec1 = ratioPrec + 1

    infinity, notANumber :: Rational
    infinity   = 1 :% 0
    notANumber = 0 :% 0

    -- Use :%, not % for Inf/NaN; the latter would
    -- immediately lead to a runtime error, because it normalises.
    \end{code}

    \begin{code}
    -- | Forms the ratio of two integral numbers.
    {-# SPECIALISE (%) :: Integer -> Integer -> Rational #-}
    (%)                     :: (Integral a) => a -> a -> Ratio a

    -- | Extract the numerator of the ratio in reduced form:
    -- the numerator and denominator have no common factor and the denominator
    -- is positive.
    numerator       :: (Integral a) => Ratio a -> a

    -- | Extract the denominator of the ratio in reduced form:
    -- the numerator and denominator have no common factor and the denominator
    -- is positive.
    denominator     :: (Integral a) => Ratio a -> a
    \end{code}

    \tr{reduce} is a subsidiary function used only in this module .
    It normalises a ratio by dividing both numerator and denominator by
    their greatest common divisor.

    \begin{code}
    reduce ::  (Integral a) => a -> a -> Ratio a
    {-# SPECIALISE reduce :: Integer -> Integer -> Rational #-}
    reduce _ 0              =  error "Ratio.%: zero denominator"
    reduce x y              =  (x `quot` d) :% (y `quot` d)
                               where d = gcd x y
    \end{code}

    \begin{code}
    x % y                   =  reduce (x * signum y) (abs y)

    numerator   (x :% _)    =  x
    denominator (_ :% y)    =  y
    \end{code}

    %*********************************************************
    %*                                                      *
    \subsection{Standard numeric classes}
    %*                                                      *
    %*********************************************************

    \begin{code}
    class  (Num a, Ord a) => Real a  where
        -- | the rational equivalent of its real argument with full precision
        toRational          ::  a -> Rational

    -- | Integral numbers, supporting integer division.
    --
    -- Minimal complete definition: 'quotRem' and 'toInteger'
    class  (Real a, Enum a) => Integral a  where
        -- | integer division truncated toward zero
        quot                :: a -> a -> a
        -- | integer remainder, satisfying
        --
        -- > (x `quot` y)*y + (x `rem` y) == x
        rem                 :: a -> a -> a
        -- | integer division truncated toward negative infinity
        div                 :: a -> a -> a
        -- | integer modulus, satisfying
        --
        -- > (x `div` y)*y + (x `mod` y) == x
        mod                 :: a -> a -> a
        -- | simultaneous 'quot' and 'rem'
        quotRem             :: a -> a -> (a,a)
        -- | simultaneous 'div' and 'mod'
        divMod              :: a -> a -> (a,a)
        -- | conversion to 'Integer'
        toInteger           :: a -> Integer

        n `quot` d          =  q  where (q,_) = quotRem n d
        n `rem` d           =  r  where (_,r) = quotRem n d
        n `div` d           =  q  where (q,_) = divMod n d
        n `mod` d           =  r  where (_,r) = divMod n d
        divMod n d          =  if signum r == negate (signum d) then (q-1, r+d) else qr
                               where qr@(q,r) = quotRem n d

    -- | Fractional numbers, supporting real division.
    --
    -- Minimal complete definition: 'fromRational' and ('recip' or @('/')@)
    class  (Num a) => Fractional a  where
        -- | fractional division
        (/)                 :: a -> a -> a
        -- | reciprocal fraction
        recip               :: a -> a
        -- | Conversion from a 'Rational' (that is @'Ratio' 'Integer'@).
        -- A floating literal stands for an application of 'fromRational'
        -- to a value of type 'Rational', so such literals have type
        -- @('Fractional' a) => a@.
        fromRational        :: Rational -> a

        recip x             =  1 / x
        x / y               = x * recip y

    -- | Extracting components of fractions.
    --
    -- Minimal complete definition: 'properFraction'
    class  (Real a, Fractional a) => RealFrac a  where
        -- | The function 'properFraction' takes a real fractional number @x@
        -- and returns a pair @(n,f)@ such that @x = n+f@, and:
        --
        -- * @n@ is an integral number with the same sign as @x@; and
        --
        -- * @f@ is a fraction with the same type and sign as @x@,
        --   and with absolute value less than @1@.
        --
        -- The default definitions of the 'ceiling', 'floor', 'truncate'
        -- and 'round' functions are in terms of 'properFraction'.
        properFraction      :: (Integral b) => a -> (b,a)
        -- | @'truncate' x@ returns the integer nearest @x@ between zero and @x@
        truncate            :: (Integral b) => a -> b
        -- | @'round' x@ returns the nearest integer to @x@
        round               :: (Integral b) => a -> b
        -- | @'ceiling' x@ returns the least integer not less than @x@
        ceiling             :: (Integral b) => a -> b
        -- | @'floor' x@ returns the greatest integer not greater than @x@
        floor               :: (Integral b) => a -> b

        truncate x          =  m  where (m,_) = properFraction x

        round x             =  let (n,r) = properFraction x
                                   m     = if r < 0 then n - 1 else n + 1
                               in case signum (abs r - 0.5) of
                                    -1 -> n
                                    0  -> if even n then n else m
                                    1  -> m

        ceiling x           =  if r > 0 then n + 1 else n
                               where (n,r) = properFraction x

        floor x             =  if r < 0 then n - 1 else n
                               where (n,r) = properFraction x
    \end{code}

    These 'numeric' enumerations come straight from the Report

    \begin{code}
    numericEnumFrom         :: (Fractional a) => a -> [a]
    numericEnumFrom         =  iterate (+1)

    numericEnumFromThen     :: (Fractional a) => a -> a -> [a]
    numericEnumFromThen n m =  iterate (+(m-n)) n

    numericEnumFromTo       :: (Ord a, Fractional a) => a -> a -> [a]
    numericEnumFromTo n m   = takeWhile (<= m + 1/2) (numericEnumFrom n)

    numericEnumFromThenTo   :: (Ord a, Fractional a) => a -> a -> a -> [a]
    numericEnumFromThenTo e1 e2 e3 = takeWhile pred (numericEnumFromThen e1 e2)
                                    where
                                     mid = (e2 - e1) / 2
                                     pred | e2 >= e1  = (<= e3 + mid)
                                          | otherwise = (>= e3 + mid)
    \end{code}

    %*********************************************************
    %*                                                      *
    \subsection{Instances for @Int@}
    %*                                                      *
    %*********************************************************

    \begin{code}
    instance  Real Int  where
        toRational x        =  toInteger x % 1

    instance  Integral Int  where
        toInteger i = int2Integer i  -- give back a full-blown Integer

        a `quot` b
         | b == 0                     = divZeroError
         | a == minBound && b == (-1) = overflowError
         | otherwise                  =  a `quotInt` b

        a `rem` b
         | b == 0                     = divZeroError
         | a == minBound && b == (-1) = overflowError
         | otherwise                  =  a `remInt` b

        a `div` b
         | b == 0                     = divZeroError
         | a == minBound && b == (-1) = overflowError
         | otherwise                  =  a `divInt` b

        a `mod` b
         | b == 0                     = divZeroError
         | a == minBound && b == (-1) = overflowError
         | otherwise                  =  a `modInt` b

        a `quotRem` b
         | b == 0                     = divZeroError
         | a == minBound && b == (-1) = overflowError
         | otherwise                  =  a `quotRemInt` b

        a `divMod` b
         | b == 0                     = divZeroError
         | a == minBound && b == (-1) = overflowError
         | otherwise                  =  a `divModInt` b
    \end{code}

    %*********************************************************
    %*                                                      *
    \subsection{Instances for @Integer@}
    %*                                                      *
    %*********************************************************

    \begin{code}
    instance  Real Integer  where
        toRational x        =  x % 1

    instance  Integral Integer where
        toInteger n      = n

        a `quot` 0 = divZeroError
        n `quot` d = n `quotInteger` d

        a `rem` 0 = divZeroError
        n `rem`  d = n `remInteger`  d

        a `divMod` 0 = divZeroError
        a `divMod` b = a `divModInteger` b

        a `quotRem` 0 = divZeroError
        a `quotRem` b = a `quotRemInteger` b

        -- use the defaults for div & mod
    \end{code}

    %*********************************************************
    %*                                                      *
    \subsection{Instances for @Ratio@}
    %*                                                      *
    %*********************************************************

    \begin{code}
    instance  (Integral a)  => Ord (Ratio a)  where
        {-# SPECIALIZE instance Ord Rational #-}
        (x:%y) <= (x':%y')  =  x * y' <= x' * y
        (x:%y) <  (x':%y')  =  x * y' <  x' * y

    instance  (Integral a)  => Num (Ratio a)  where
        {-# SPECIALIZE instance Num Rational #-}
        (x:%y) + (x':%y')   =  reduce (x*y' + x'*y) (y*y')
        (x:%y) - (x':%y')   =  reduce (x*y' - x'*y) (y*y')
        (x:%y) * (x':%y')   =  reduce (x * x') (y * y')
        negate (x:%y)       =  (-x) :% y
        abs (x:%y)          =  abs x :% y
        signum (x:%_)       =  signum x :% 1
        fromInteger x       =  fromInteger x :% 1

    instance  (Integral a)  => Fractional (Ratio a)  where
        {-# SPECIALIZE instance Fractional Rational #-}
        (x:%y) / (x':%y')   =  (x*y') % (y*x')
        recip (x:%y)        =  y % x
        fromRational (x:%y) =  fromInteger x :% fromInteger y

    instance  (Integral a)  => Real (Ratio a)  where
        {-# SPECIALIZE instance Real Rational #-}
        toRational (x:%y)   =  toInteger x :% toInteger y

    instance  (Integral a)  => RealFrac (Ratio a)  where
        {-# SPECIALIZE instance RealFrac Rational #-}
        properFraction (x:%y) = (fromInteger (toInteger q), r:%y)
                              where (q,r) = quotRem x y

    instance  (Integral a)  => Show (Ratio a)  where
        {-# SPECIALIZE instance Show Rational #-}
        showsPrec p (x:%y)  =  showParen (p > ratioPrec) $
                               showsPrec ratioPrec1 x .
                               showString "%" .     -- H98 report has spaces round the %
                                                    -- but we removed them [May 04]
                               showsPrec ratioPrec1 y

    instance  (Integral a)  => Enum (Ratio a)  where
        {-# SPECIALIZE instance Enum Rational #-}
        succ x              =  x + 1
        pred x              =  x - 1

        toEnum n            =  fromInteger (int2Integer n) :% 1
        fromEnum            =  fromInteger . truncate

        enumFrom            =  numericEnumFrom
        enumFromThen        =  numericEnumFromThen
        enumFromTo          =  numericEnumFromTo
        enumFromThenTo      =  numericEnumFromThenTo
    \end{code}

    %*********************************************************
    %*                                                      *
    \subsection{Coercions}
    %*                                                      *
    %*********************************************************

    \begin{code}
    -- | general coercion from integral types
    fromIntegral :: (Integral a, Num b) => a -> b
    fromIntegral = fromInteger . toInteger

    {-# RULES
    "fromIntegral/Int->Int" fromIntegral = id :: Int -> Int
        #-}

    -- | general coercion to fractional types
    realToFrac :: (Real a, Fractional b) => a -> b
    realToFrac = fromRational . toRational

    {-# RULES
    "realToFrac/Int->Int" realToFrac = id :: Int -> Int
        #-}
    \end{code}

    %*********************************************************
    %*                                                      *
    \subsection{Overloaded numeric functions}
    %*                                                      *
    %*********************************************************

    \begin{code}
    -- | Converts a possibly-negative 'Real' value to a string.
    showSigned :: (Real a)
      => (a -> ShowS)       -- ^ a function that can show unsigned values
      -> Int                -- ^ the precedence of the enclosing context
      -> a                  -- ^ the value to show
      -> ShowS
    showSigned showPos p x
       | x < 0     = showParen (p > 6) (showChar '-' . showPos (-x))
       | otherwise = showPos x

    even, odd       :: (Integral a) => a -> Bool
    even n          =  n `rem` 2 == 0
    odd             =  not . even

    -------------------------------------------------------
    -- | raise a number to a non-negative integral power
    {-# SPECIALISE (^) ::
            Integer -> Integer -> Integer,
            Integer -> Int -> Integer,
            Int -> Int -> Int #-}
    (^) :: (Num a, Integral b) => a -> b -> a
    x0 ^ y0 | y0 < 0    = error "Negative exponent"
            | y0 == 0   = 1
            | otherwise = f x0 y0
        where -- f : x0 ^ y0 = x ^ y
              f x y | even y    = f (x * x) (y `quot` 2)
                    | y == 1    = x
                    | otherwise = g (x * x) ((y - 1) `quot` 2) x
              -- g : x0 ^ y0 = (x ^ y) * z
              g x y z | even y = g (x * x) (y `quot` 2) z
                      | y == 1 = x * z
                      | otherwise = g (x * x) ((y - 1) `quot` 2) (x * z)

    -- | raise a number to an integral power
    {-# SPECIALISE (^^) ::
            Rational -> Int -> Rational #-}
    (^^)            :: (Fractional a, Integral b) => a -> b -> a
    x ^^ n          =  if n >= 0 then x^n else recip (x^(negate n))

    -------------------------------------------------------
    -- | @'gcd' x y@ is the greatest (positive) integer that divides both @x@
    -- and @y@; for example @'gcd' (-3) 6@ = @3@, @'gcd' (-3) (-6)@ = @3@,
    -- @'gcd' 0 4@ = @4@.  @'gcd' 0 0@ raises a runtime error.
    gcd             :: (Integral a) => a -> a -> a
    gcd 0 0         =  error "Prelude.gcd: gcd 0 0 is undefined"
    gcd x y         =  gcd' (abs x) (abs y)
                       where gcd' a 0  =  a
                             gcd' a b  =  gcd' b (a `rem` b)

    -- | @'lcm' x y@ is the smallest positive integer that both @x@ and @y@ divide.
    lcm             :: (Integral a) => a -> a -> a
    {-# SPECIALISE lcm :: Int -> Int -> Int #-}
    lcm _ 0         =  0
    lcm 0 _         =  0
    lcm x y         =  abs ((x `quot` (gcd x y)) * y)

    {-# RULES
    "gcd/Int->Int->Int"             gcd = gcdInt
    "gcd/Integer->Integer->Integer" gcd = gcdInteger
    "lcm/Integer->Integer->Integer" lcm = lcmInteger
     #-}

    integralEnumFrom :: (Integral a, Bounded a) => a -> [a]
    integralEnumFrom n = map fromInteger [toInteger n .. toInteger (maxBound `asTypeOf` n)]

    integralEnumFromThen :: (Integral a, Bounded a) => a -> a -> [a]
    integralEnumFromThen n1 n2
      | i_n2 >= i_n1  = map fromInteger [i_n1, i_n2 .. toInteger (maxBound `asTypeOf` n1)]
      | otherwise     = map fromInteger [i_n1, i_n2 .. toInteger (minBound `asTypeOf` n1)]
      where
        i_n1 = toInteger n1
        i_n2 = toInteger n2

    integralEnumFromTo :: Integral a => a -> a -> [a]
    integralEnumFromTo n m = map fromInteger [toInteger n .. toInteger m]

    integralEnumFromThenTo :: Integral a => a -> a -> a -> [a]
    integralEnumFromThenTo n1 n2 m
      = map fromInteger [toInteger n1, toInteger n2 .. toInteger m]
    \end{code}