1 % ------------------------------------------------------------------------------
2 % $Id: PrelReal.lhs,v 1.13 2001/03/29 08:03:47 qrczak Exp $
4 % (c) The University of Glasgow, 1994-2000
7 \section[PrelReal]{Module @PrelReal@}
22 {-# OPTIONS -fno-implicit-prelude #-}
26 import {-# SOURCE #-} PrelErr
34 infixl 7 /, `quot`, `rem`, `div`, `mod`
36 default () -- Double isn't available yet,
37 -- and we shouldn't be using defaults anyway
41 %*********************************************************
43 \subsection{The @Ratio@ and @Rational@ types}
45 %*********************************************************
48 data (Integral a) => Ratio a = !a :% !a deriving (Eq)
49 type Rational = Ratio Integer
54 {-# SPECIALISE (%) :: Integer -> Integer -> Rational #-}
55 (%) :: (Integral a) => a -> a -> Ratio a
56 numerator, denominator :: (Integral a) => Ratio a -> a
59 \tr{reduce} is a subsidiary function used only in this module .
60 It normalises a ratio by dividing both numerator and denominator by
61 their greatest common divisor.
64 reduce :: (Integral a) => a -> a -> Ratio a
65 reduce _ 0 = error "Ratio.%: zero denominator"
66 reduce x y = (x `quot` d) :% (y `quot` d)
71 x % y = reduce (x * signum y) (abs y)
73 numerator (x :% _) = x
74 denominator (_ :% y) = y
78 %*********************************************************
80 \subsection{Standard numeric classes}
82 %*********************************************************
85 class (Num a, Ord a) => Real a where
86 toRational :: a -> Rational
88 class (Real a, Enum a) => Integral a where
89 quot, rem, div, mod :: a -> a -> a
90 quotRem, divMod :: a -> a -> (a,a)
91 toInteger :: a -> Integer
93 n `quot` d = q where (q,_) = quotRem n d
94 n `rem` d = r where (_,r) = quotRem n d
95 n `div` d = q where (q,_) = divMod n d
96 n `mod` d = r where (_,r) = divMod n d
97 divMod n d = if signum r == negate (signum d) then (q-1, r+d) else qr
98 where qr@(q,r) = quotRem n d
100 class (Num a) => Fractional a where
103 fromRational :: Rational -> a
108 class (Real a, Fractional a) => RealFrac a where
109 properFraction :: (Integral b) => a -> (b,a)
110 truncate, round :: (Integral b) => a -> b
111 ceiling, floor :: (Integral b) => a -> b
113 truncate x = m where (m,_) = properFraction x
115 round x = let (n,r) = properFraction x
116 m = if r < 0 then n - 1 else n + 1
117 in case signum (abs r - 0.5) of
119 0 -> if even n then n else m
122 ceiling x = if r > 0 then n + 1 else n
123 where (n,r) = properFraction x
125 floor x = if r < 0 then n - 1 else n
126 where (n,r) = properFraction x
130 These 'numeric' enumerations come straight from the Report
133 numericEnumFrom :: (Fractional a) => a -> [a]
134 numericEnumFrom = iterate (+1)
136 numericEnumFromThen :: (Fractional a) => a -> a -> [a]
137 numericEnumFromThen n m = iterate (+(m-n)) n
139 numericEnumFromTo :: (Ord a, Fractional a) => a -> a -> [a]
140 numericEnumFromTo n m = takeWhile (<= m + 1/2) (numericEnumFrom n)
142 numericEnumFromThenTo :: (Ord a, Fractional a) => a -> a -> a -> [a]
143 numericEnumFromThenTo e1 e2 e3 = takeWhile pred (numericEnumFromThen e1 e2)
146 pred | e2 > e1 = (<= e3 + mid)
147 | otherwise = (>= e3 + mid)
151 %*********************************************************
153 \subsection{Instances for @Int@}
155 %*********************************************************
158 instance Real Int where
159 toRational x = toInteger x % 1
161 instance Integral Int where
162 toInteger i = int2Integer i -- give back a full-blown Integer
164 -- Following chks for zero divisor are non-standard (WDP)
165 a `quot` b = if b /= 0
167 else error "Prelude.Integral.quot{Int}: divide by 0"
168 a `rem` b = if b /= 0
170 else error "Prelude.Integral.rem{Int}: divide by 0"
172 x `div` y = x `divInt` y
173 x `mod` y = x `modInt` y
175 a `quotRem` b = a `quotRemInt` b
176 a `divMod` b = a `divModInt` b
180 %*********************************************************
182 \subsection{Instances for @Integer@}
184 %*********************************************************
187 instance Real Integer where
190 instance Integral Integer where
193 n `quot` d = n `quotInteger` d
194 n `rem` d = n `remInteger` d
196 n `div` d = q where (q,_) = divMod n d
197 n `mod` d = r where (_,r) = divMod n d
199 a `divMod` b = a `divModInteger` b
200 a `quotRem` b = a `quotRemInteger` b
204 %*********************************************************
206 \subsection{Instances for @Ratio@}
208 %*********************************************************
211 instance (Integral a) => Ord (Ratio a) where
212 {-# SPECIALIZE instance Ord Rational #-}
213 (x:%y) <= (x':%y') = x * y' <= x' * y
214 (x:%y) < (x':%y') = x * y' < x' * y
216 instance (Integral a) => Num (Ratio a) where
217 {-# SPECIALIZE instance Num Rational #-}
218 (x:%y) + (x':%y') = reduce (x*y' + x'*y) (y*y')
219 (x:%y) - (x':%y') = reduce (x*y' - x'*y) (y*y')
220 (x:%y) * (x':%y') = reduce (x * x') (y * y')
221 negate (x:%y) = (-x) :% y
222 abs (x:%y) = abs x :% y
223 signum (x:%_) = signum x :% 1
224 fromInteger x = fromInteger x :% 1
226 instance (Integral a) => Fractional (Ratio a) where
227 {-# SPECIALIZE instance Fractional Rational #-}
228 (x:%y) / (x':%y') = (x*y') % (y*x')
229 recip (x:%y) = if x < 0 then (-y) :% (-x) else y :% x
230 fromRational (x:%y) = fromInteger x :% fromInteger y
232 instance (Integral a) => Real (Ratio a) where
233 {-# SPECIALIZE instance Real Rational #-}
234 toRational (x:%y) = toInteger x :% toInteger y
236 instance (Integral a) => RealFrac (Ratio a) where
237 {-# SPECIALIZE instance RealFrac Rational #-}
238 properFraction (x:%y) = (fromInteger (toInteger q), r:%y)
239 where (q,r) = quotRem x y
241 instance (Integral a) => Show (Ratio a) where
242 {-# SPECIALIZE instance Show Rational #-}
243 showsPrec p (x:%y) = showParen (p > ratio_prec)
244 (shows x . showString " % " . shows y)
249 instance (Integral a) => Enum (Ratio a) where
250 {-# SPECIALIZE instance Enum Rational #-}
254 toEnum n = fromInteger (int2Integer n) :% 1
255 fromEnum = fromInteger . truncate
257 enumFrom = numericEnumFrom
258 enumFromThen = numericEnumFromThen
259 enumFromTo = numericEnumFromTo
260 enumFromThenTo = numericEnumFromThenTo
264 %*********************************************************
266 \subsection{Coercions}
268 %*********************************************************
271 fromIntegral :: (Integral a, Num b) => a -> b
272 fromIntegral = fromInteger . toInteger
275 "fromIntegral/Int->Int" fromIntegral = id :: Int -> Int
278 realToFrac :: (Real a, Fractional b) => a -> b
279 realToFrac = fromRational . toRational
282 "realToFrac/Int->Int" realToFrac = id :: Int -> Int
285 -- For backward compatibility
286 {-# DEPRECATED fromInt "use fromIntegral instead" #-}
287 fromInt :: Num a => Int -> a
288 fromInt = fromIntegral
290 -- For backward compatibility
291 {-# DEPRECATED toInt "use fromIntegral instead" #-}
292 toInt :: Integral a => a -> Int
296 %*********************************************************
298 \subsection{Overloaded numeric functions}
300 %*********************************************************
303 showSigned :: (Real a) => (a -> ShowS) -> Int -> a -> ShowS
304 showSigned showPos p x
305 | x < 0 = showParen (p > 6) (showChar '-' . showPos (-x))
306 | otherwise = showPos x
308 even, odd :: (Integral a) => a -> Bool
309 even n = n `rem` 2 == 0
312 -------------------------------------------------------
313 {-# SPECIALISE (^) ::
314 Integer -> Integer -> Integer,
315 Integer -> Int -> Integer,
316 Int -> Int -> Int #-}
317 (^) :: (Num a, Integral b) => a -> b -> a
319 x ^ n | n > 0 = f x (n-1) x
321 f a d y = g a d where
322 g b i | even i = g (b*b) (i `quot` 2)
323 | otherwise = f b (i-1) (b*y)
324 _ ^ _ = error "Prelude.^: negative exponent"
326 {-# SPECIALISE (^^) ::
327 Rational -> Int -> Rational #-}
328 (^^) :: (Fractional a, Integral b) => a -> b -> a
329 x ^^ n = if n >= 0 then x^n else recip (x^(negate n))
332 -------------------------------------------------------
333 gcd :: (Integral a) => a -> a -> a
334 gcd 0 0 = error "Prelude.gcd: gcd 0 0 is undefined"
335 gcd x y = gcd' (abs x) (abs y)
337 gcd' a b = gcd' b (a `rem` b)
339 lcm :: (Integral a) => a -> a -> a
340 {-# SPECIALISE lcm :: Int -> Int -> Int #-}
343 lcm x y = abs ((x `quot` (gcd x y)) * y)
347 "gcd/Int->Int->Int" gcd = gcdInt
348 "gcd/Integer->Integer->Integer" gcd = gcdInteger
349 "lcm/Integer->Integer->Integer" lcm = lcmInteger
352 integralEnumFrom :: (Integral a, Bounded a) => a -> [a]
353 integralEnumFrom n = map fromInteger [toInteger n .. toInteger (maxBound `asTypeOf` n)]
355 integralEnumFromThen :: (Integral a, Bounded a) => a -> a -> [a]
356 integralEnumFromThen n1 n2
357 | i_n2 >= i_n1 = map fromInteger [i_n1, i_n2 .. toInteger (maxBound `asTypeOf` n1)]
358 | otherwise = map fromInteger [i_n1, i_n2 .. toInteger (minBound `asTypeOf` n1)]
363 integralEnumFromTo :: Integral a => a -> a -> [a]
364 integralEnumFromTo n m = map fromInteger [toInteger n .. toInteger m]
366 integralEnumFromThenTo :: Integral a => a -> a -> a -> [a]
367 integralEnumFromThenTo n1 n2 m
368 = map fromInteger [toInteger n1, toInteger n2 .. toInteger m]