```\begin{code}
{-# OPTIONS_GHC -XNoImplicitPrelude #-}
-----------------------------------------------------------------------------
-- |
-- Module      :  GHC.Real
-- Copyright   :  (c) The University of Glasgow, 1994-2002
--
-- 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

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@;
--   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

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

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]
-- 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

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}

%*********************************************************
%*                                                      *
%*                                                      *
%*********************************************************

\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)

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

-- XXX to use another Integer implementation, you might need to disable
-- the gcd/Integer and lcm/Integer RULES above
--
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}
```