\begin{code}

    {-# OPTIONS_GHC -XNoImplicitPrelude #-}
    {-# OPTIONS_HADDOCK hide #-}
    -----------------------------------------------------------------------------
    -- |
    -- Module      :  GHC.Real
    -- Copyright   :  (c) The University of Glasgow, 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
    import GHC.Err

    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

        {-# INLINE quot #-}
        {-# INLINE rem #-}
        {-# INLINE div #-}
        {-# INLINE mod #-}
        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

        {-# INLINE recip #-}
        {-# INLINE (/) #-}
        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@;
        --   the even integer if @x@ is equidistant between two integers
        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

        {-# INLINE truncate #-}
        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
                                    _  -> error "round default defn: Bad value"

        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 n	=  n `seq` (n : numericEnumFrom (n + 1))

    numericEnumFromThen     :: (Fractional a) => a -> a -> [a]
    numericEnumFromThen n m	= n `seq` m `seq` (n : numericEnumFromThen m (m+m-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 predicate (numericEnumFromThen e1 e2)
                                    where
                                     mid = (e2 - e1) / 2
                                     predicate | 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# i) = smallInteger i

        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

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

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

        _ `divMod` 0 = divZeroError
        a `divMod` b = case a `divModInteger` b of
                       (# x, y #) -> (x, y)

        _ `quotRem` 0 = divZeroError
        a `quotRem` b = case a `quotRemInteger` b of
                        (# q, r #) -> (q, r)

        -- 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

    {-# RULES "fromRational/id" fromRational = id :: Rational -> Rational #-}
    instance  (Integral a)  => Fractional (Ratio a)  where
        {-# SPECIALIZE instance Fractional Rational #-}
        (x:%y) / (x':%y')   =  (x*y') % (y*x')
        recip (0:%_)        = error "Ratio.%: zero denominator"
        recip (x:%y)
            | x < 0         = negate y :% negate x
            | otherwise     = 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]
                               -- and added them again for consistency with
                               -- Haskell 98 [Sep 08, #1920]
                               showsPrec ratioPrec1 y

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

        toEnum n            =  fromIntegral 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 #-}
    {-# INLINABLE (^) #-}    -- See Note [Inlining (^)]
    (^) :: (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
    (^^)            :: (Fractional a, Integral b) => a -> b -> a
    {-# INLINABLE (^^) #-}         -- See Note [Inlining (^)
    x ^^ n          =  if n >= 0 then x^n else recip (x^(negate n))

    {- Note [Inlining (^)
       ~~~~~~~~~~~~~~~~~~~~~
       The INLINABLE pragma allows (^) to be specialised at its call sites.
       If it is called repeatedly at the same type, that can make a huge
       difference, because of those constants which can be repeatedly
       calculated.

       Currently the fromInteger calls are not floated because we get
                 \d1 d2 x y -> blah
       after the gentle round of simplification. -}

    -------------------------------------------------------
    -- Special power functions for Rational
    --
    -- see #4337
    --
    -- Rationale:
    -- For a legitimate Rational (n :% d), the numerator and denominator are
    -- coprime, i.e. they have no common prime factor.
    -- Therefore all powers (n ^ a) and (d ^ b) are also coprime, so it is
    -- not necessary to compute the greatest common divisor, which would be
    -- done in the default implementation at each multiplication step.
    -- Since exponentiation quickly leads to very large numbers and
    -- calculation of gcds is generally very slow for large numbers,
    -- avoiding the gcd leads to an order of magnitude speedup relatively
    -- soon (and an asymptotic improvement overall).
    --
    -- Note:
    -- We cannot use these functions for general Ratio a because that would
    -- change results in a multitude of cases.
    -- The cause is that if a and b are coprime, their remainders by any
    -- positive modulus generally aren't, so in the default implementation
    -- reduction occurs.
    --
    -- Example:
    -- (17 % 3) ^ 3 :: Ratio Word8
    -- Default:
    -- (17 % 3) ^ 3 = ((17 % 3) ^ 2) * (17 % 3)
    --              = ((289 `mod` 256) % 9) * (17 % 3)
    --              = (33 % 9) * (17 % 3)
    --              = (11 % 3) * (17 % 3)
    --              = (187 % 9)
    -- But:
    -- ((17^3) `mod` 256) % (3^3)   = (4913 `mod` 256) % 27
    --                              = 49 % 27
    --
    -- TODO:
    -- Find out whether special-casing for numerator, denominator or
    -- exponent = 1 (or -1, where that may apply) gains something.

    -- Special version of (^) for Rational base
    {-# RULES "(^)/Rational"    (^) = (^%^) #-}
    (^%^)           :: Integral a => Rational -> a -> Rational
    (n :% d) ^%^ e
        | e < 0     = error "Negative exponent"
        | e == 0    = 1 :% 1
        | otherwise = (n ^ e) :% (d ^ e)

    -- Special version of (^^) for Rational base
    {-# RULES "(^^)/Rational"   (^^) = (^^%^^) #-}
    (^^%^^)         :: Integral a => Rational -> a -> Rational
    (n :% d) ^^%^^ e
        | e > 0     = (n ^ e) :% (d ^ e)
        | e == 0    = 1 :% 1
        | n > 0     = (d ^ (negate e)) :% (n ^ (negate e))
        | n == 0    = error "Ratio.%: zero denominator"
        | otherwise = let nn = d ^ (negate e)
                          dd = (negate n) ^ (negate e)
                      in if even e then (nn :% dd) else (negate nn :% dd)

    -------------------------------------------------------
    -- | @'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)

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

    gcdInteger' :: Integer -> Integer -> Integer
    gcdInteger' 0 0 = error "GHC.Real.gcdInteger': gcd 0 0 is undefined"
    gcdInteger' a b = gcdInteger a b

    gcdInt :: Int -> Int -> Int
    gcdInt 0 0 = error "GHC.Real.gcdInt: gcd 0 0 is undefined"
    gcdInt a b = fromIntegral (gcdInteger (fromIntegral a) (fromIntegral b))
    #endif

    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}