\begin{code}
{-# OPTIONS_GHC -fno-implicit-prelude #-}
-----------------------------------------------------------------------------
-- |
-- Module : GHC.Real
-- Copyright : (c) The FFI Task Force, 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 {-# SOURCE #-} GHC.Err
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
_ ^ 0 = 1
x ^ n | n > 0 = f x (n-1) x
where f _ 0 y = y
f a d y = g a d where
g b i | even i = g (b*b) (i `quot` 2)
| otherwise = f b (i-1) (b*y)
_ ^ _ = error "Prelude.^: negative exponent"
-- | 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}
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