+++ /dev/null
-% ------------------------------------------------------------------------------
-% $Id: PrelReal.lhs,v 1.16 2001/09/26 16:27:04 simonpj Exp $
-%
-% (c) The University of Glasgow, 1994-2000
-%
-
-\section[PrelReal]{Module @PrelReal@}
-
-The types
-
- Ratio, Rational
-
-and the classes
-
- Real
- Integral
- Fractional
- RealFrac
-
-
-\begin{code}
-{-# OPTIONS -fno-implicit-prelude #-}
-
-module PrelReal where
-
-import {-# SOURCE #-} PrelErr
-import PrelBase
-import PrelNum
-import PrelList
-import PrelEnum
-import PrelShow
-
-infixr 8 ^, ^^
-infixl 7 /, `quot`, `rem`, `div`, `mod`
-
-default () -- Double isn't available yet,
- -- and we shouldn't be using defaults anyway
-\end{code}
-
-
-%*********************************************************
-%* *
-\subsection{The @Ratio@ and @Rational@ types}
-%* *
-%*********************************************************
-
-\begin{code}
-data (Integral a) => Ratio a = !a :% !a deriving (Eq)
-type Rational = Ratio Integer
-\end{code}
-
-
-\begin{code}
-{-# SPECIALISE (%) :: Integer -> Integer -> Rational #-}
-(%) :: (Integral a) => a -> a -> Ratio a
-numerator, 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
- toRational :: a -> Rational
-
-class (Real a, Enum a) => Integral a where
- quot, rem, div, mod :: a -> a -> a
- quotRem, divMod :: a -> a -> (a,a)
- 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
-
-class (Num a) => Fractional a where
- (/) :: a -> a -> a
- recip :: a -> a
- fromRational :: Rational -> a
-
- recip x = 1 / x
- x / y = x * recip y
-
-class (Real a, Fractional a) => RealFrac a where
- properFraction :: (Integral b) => a -> (b,a)
- truncate, round :: (Integral b) => a -> b
- ceiling, 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
-
- -- Following chks for zero divisor are non-standard (WDP)
- a `quot` b = if b /= 0
- then a `quotInt` b
- else error "Prelude.Integral.quot{Int}: divide by 0"
- a `rem` b = if b /= 0
- then a `remInt` b
- else error "Prelude.Integral.rem{Int}: divide by 0"
-
- x `div` y = x `divInt` y
- x `mod` y = x `modInt` y
-
- a `quotRem` b = a `quotRemInt` b
- a `divMod` b = 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
-
- n `quot` d = n `quotInteger` d
- n `rem` d = n `remInteger` d
-
- n `div` d = q where (q,_) = divMod n d
- n `mod` d = r where (_,r) = divMod n d
-
- a `divMod` b = a `divModInteger` b
- a `quotRem` b = a `quotRemInteger` b
-\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 > ratio_prec)
- (shows x . showString " % " . shows y)
-
-ratio_prec :: Int
-ratio_prec = 7
-
-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}
-fromIntegral :: (Integral a, Num b) => a -> b
-fromIntegral = fromInteger . toInteger
-
-{-# RULES
-"fromIntegral/Int->Int" fromIntegral = id :: Int -> Int
- #-}
-
-realToFrac :: (Real a, Fractional b) => a -> b
-realToFrac = fromRational . toRational
-
-{-# RULES
-"realToFrac/Int->Int" realToFrac = id :: Int -> Int
- #-}
-
--- For backward compatibility
-{-# DEPRECATED fromInt "use fromIntegral instead" #-}
-fromInt :: Num a => Int -> a
-fromInt = fromIntegral
-
--- For backward compatibility
-{-# DEPRECATED toInt "use fromIntegral instead" #-}
-toInt :: Integral a => a -> Int
-toInt = fromIntegral
-\end{code}
-
-%*********************************************************
-%* *
-\subsection{Overloaded numeric functions}
-%* *
-%*********************************************************
-
-\begin{code}
-showSigned :: (Real a) => (a -> ShowS) -> Int -> a -> 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
-
--------------------------------------------------------
-{-# 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"
-
-{-# 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 :: (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 :: (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}
projects / ghc-hetmet.git / blobdiff