\begin{code}
{-# LANGUAGE Trustworthy #-}
{-# LANGUAGE CPP, NoImplicitPrelude, MagicHash, UnboxedTuples #-}
{-# OPTIONS_HADDOCK hide #-}
-----------------------------------------------------------------------------
-- |
-- Module      :  GHC.Real
-- Copyright   :  (c) The University of Glasgow, 1994-2002
-- License     :  see libraries/base/LICENSE
--
-- Maintainer  :  [email protected]
-- 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  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
     | b == (-1) && a == minBound = overflowError -- Note [Order of tests]
                                                  -- in GHC.Int
     | otherwise                  =  a `quotInt` b

    a `rem` b
     | b == 0                     = divZeroError
       -- The quotRem CPU instruction fails for minBound `quotRem` -1,
       -- but minBound `rem` -1 is well-defined (0). We therefore
       -- special-case it.
     | b == (-1)                  = 0
     | otherwise                  =  a `remInt` b

    a `div` b
     | b == 0                     = divZeroError
     | b == (-1) && a == minBound = overflowError -- Note [Order of tests]
                                                  -- in GHC.Int
     | otherwise                  =  a `divInt` b

    a `mod` b
     | b == 0                     = divZeroError
       -- The divMod CPU instruction fails for minBound `divMod` -1,
       -- but minBound `mod` -1 is well-defined (0). We therefore
       -- special-case it.
     | b == (-1)                  = 0
     | otherwise                  =  a `modInt` b

    a `quotRem` b
     | b == 0                     = divZeroError
       -- Note [Order of tests] in GHC.Int
     | b == (-1) && a == minBound = (overflowError, 0)
     | otherwise                  =  a `quotRemInt` b

    a `divMod` b
     | b == 0                     = divZeroError
       -- Note [Order of tests] in GHC.Int
     | b == (-1) && a == minBound = (overflowError, 0)
     | 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 non-negative factor of both @x@ and @y@ of which
-- every common factor of @x@ and @y@ is also a factor; for example
-- @'gcd' 4 2 = 2@, @'gcd' (-4) 6 = 2@, @'gcd' 0 4@ = @4@. @'gcd' 0 0@ = @0@.
-- (That is, the common divisor that is \"greatest\" in the divisibility
-- preordering.)
--
-- Note: Since for signed fixed-width integer types, @'abs' 'minBound' < 0@,
-- the result may be negative if one of the arguments is @'minBound'@ (and
-- necessarily is if the other is @0@ or @'minBound'@) for such types.
gcd             :: (Integral a) => a -> a -> a
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
 #-}

gcdInt :: Int -> Int -> Int
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}