1 % ------------------------------------------------------------------------------
2 % $Id: PrelReal.lhs,v 1.16 2001/09/26 16:27:04 simonpj 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 {-# SPECIALISE reduce :: Integer -> Integer -> Rational #-}
66 reduce _ 0 = error "Ratio.%: zero denominator"
67 reduce x y = (x `quot` d) :% (y `quot` d)
72 x % y = reduce (x * signum y) (abs y)
74 numerator (x :% _) = x
75 denominator (_ :% y) = y
79 %*********************************************************
81 \subsection{Standard numeric classes}
83 %*********************************************************
86 class (Num a, Ord a) => Real a where
87 toRational :: a -> Rational
89 class (Real a, Enum a) => Integral a where
90 quot, rem, div, mod :: a -> a -> a
91 quotRem, divMod :: a -> a -> (a,a)
92 toInteger :: a -> Integer
94 n `quot` d = q where (q,_) = quotRem n d
95 n `rem` d = r where (_,r) = quotRem n d
96 n `div` d = q where (q,_) = divMod n d
97 n `mod` d = r where (_,r) = divMod n d
98 divMod n d = if signum r == negate (signum d) then (q-1, r+d) else qr
99 where qr@(q,r) = quotRem n d
101 class (Num a) => Fractional a where
104 fromRational :: Rational -> a
109 class (Real a, Fractional a) => RealFrac a where
110 properFraction :: (Integral b) => a -> (b,a)
111 truncate, round :: (Integral b) => a -> b
112 ceiling, floor :: (Integral b) => a -> b
114 truncate x = m where (m,_) = properFraction x
116 round x = let (n,r) = properFraction x
117 m = if r < 0 then n - 1 else n + 1
118 in case signum (abs r - 0.5) of
120 0 -> if even n then n else m
123 ceiling x = if r > 0 then n + 1 else n
124 where (n,r) = properFraction x
126 floor x = if r < 0 then n - 1 else n
127 where (n,r) = properFraction x
131 These 'numeric' enumerations come straight from the Report
134 numericEnumFrom :: (Fractional a) => a -> [a]
135 numericEnumFrom = iterate (+1)
137 numericEnumFromThen :: (Fractional a) => a -> a -> [a]
138 numericEnumFromThen n m = iterate (+(m-n)) n
140 numericEnumFromTo :: (Ord a, Fractional a) => a -> a -> [a]
141 numericEnumFromTo n m = takeWhile (<= m + 1/2) (numericEnumFrom n)
143 numericEnumFromThenTo :: (Ord a, Fractional a) => a -> a -> a -> [a]
144 numericEnumFromThenTo e1 e2 e3 = takeWhile pred (numericEnumFromThen e1 e2)
147 pred | e2 > e1 = (<= e3 + mid)
148 | otherwise = (>= e3 + mid)
152 %*********************************************************
154 \subsection{Instances for @Int@}
156 %*********************************************************
159 instance Real Int where
160 toRational x = toInteger x % 1
162 instance Integral Int where
163 toInteger i = int2Integer i -- give back a full-blown Integer
165 -- Following chks for zero divisor are non-standard (WDP)
166 a `quot` b = if b /= 0
168 else error "Prelude.Integral.quot{Int}: divide by 0"
169 a `rem` b = if b /= 0
171 else error "Prelude.Integral.rem{Int}: divide by 0"
173 x `div` y = x `divInt` y
174 x `mod` y = x `modInt` y
176 a `quotRem` b = a `quotRemInt` b
177 a `divMod` b = a `divModInt` b
181 %*********************************************************
183 \subsection{Instances for @Integer@}
185 %*********************************************************
188 instance Real Integer where
191 instance Integral Integer where
194 n `quot` d = n `quotInteger` d
195 n `rem` d = n `remInteger` d
197 n `div` d = q where (q,_) = divMod n d
198 n `mod` d = r where (_,r) = divMod n d
200 a `divMod` b = a `divModInteger` b
201 a `quotRem` b = a `quotRemInteger` b
205 %*********************************************************
207 \subsection{Instances for @Ratio@}
209 %*********************************************************
212 instance (Integral a) => Ord (Ratio a) where
213 {-# SPECIALIZE instance Ord Rational #-}
214 (x:%y) <= (x':%y') = x * y' <= x' * y
215 (x:%y) < (x':%y') = x * y' < x' * y
217 instance (Integral a) => Num (Ratio a) where
218 {-# SPECIALIZE instance Num Rational #-}
219 (x:%y) + (x':%y') = reduce (x*y' + x'*y) (y*y')
220 (x:%y) - (x':%y') = reduce (x*y' - x'*y) (y*y')
221 (x:%y) * (x':%y') = reduce (x * x') (y * y')
222 negate (x:%y) = (-x) :% y
223 abs (x:%y) = abs x :% y
224 signum (x:%_) = signum x :% 1
225 fromInteger x = fromInteger x :% 1
227 instance (Integral a) => Fractional (Ratio a) where
228 {-# SPECIALIZE instance Fractional Rational #-}
229 (x:%y) / (x':%y') = (x*y') % (y*x')
231 fromRational (x:%y) = fromInteger x :% fromInteger y
233 instance (Integral a) => Real (Ratio a) where
234 {-# SPECIALIZE instance Real Rational #-}
235 toRational (x:%y) = toInteger x :% toInteger y
237 instance (Integral a) => RealFrac (Ratio a) where
238 {-# SPECIALIZE instance RealFrac Rational #-}
239 properFraction (x:%y) = (fromInteger (toInteger q), r:%y)
240 where (q,r) = quotRem x y
242 instance (Integral a) => Show (Ratio a) where
243 {-# SPECIALIZE instance Show Rational #-}
244 showsPrec p (x:%y) = showParen (p > ratio_prec)
245 (shows x . showString " % " . shows y)
250 instance (Integral a) => Enum (Ratio a) where
251 {-# SPECIALIZE instance Enum Rational #-}
255 toEnum n = fromInteger (int2Integer n) :% 1
256 fromEnum = fromInteger . truncate
258 enumFrom = numericEnumFrom
259 enumFromThen = numericEnumFromThen
260 enumFromTo = numericEnumFromTo
261 enumFromThenTo = numericEnumFromThenTo
265 %*********************************************************
267 \subsection{Coercions}
269 %*********************************************************
272 fromIntegral :: (Integral a, Num b) => a -> b
273 fromIntegral = fromInteger . toInteger
276 "fromIntegral/Int->Int" fromIntegral = id :: Int -> Int
279 realToFrac :: (Real a, Fractional b) => a -> b
280 realToFrac = fromRational . toRational
283 "realToFrac/Int->Int" realToFrac = id :: Int -> Int
286 -- For backward compatibility
287 {-# DEPRECATED fromInt "use fromIntegral instead" #-}
288 fromInt :: Num a => Int -> a
289 fromInt = fromIntegral
291 -- For backward compatibility
292 {-# DEPRECATED toInt "use fromIntegral instead" #-}
293 toInt :: Integral a => a -> Int
297 %*********************************************************
299 \subsection{Overloaded numeric functions}
301 %*********************************************************
304 showSigned :: (Real a) => (a -> ShowS) -> Int -> a -> ShowS
305 showSigned showPos p x
306 | x < 0 = showParen (p > 6) (showChar '-' . showPos (-x))
307 | otherwise = showPos x
309 even, odd :: (Integral a) => a -> Bool
310 even n = n `rem` 2 == 0
313 -------------------------------------------------------
314 {-# SPECIALISE (^) ::
315 Integer -> Integer -> Integer,
316 Integer -> Int -> Integer,
317 Int -> Int -> Int #-}
318 (^) :: (Num a, Integral b) => a -> b -> a
320 x ^ n | n > 0 = f x (n-1) x
322 f a d y = g a d where
323 g b i | even i = g (b*b) (i `quot` 2)
324 | otherwise = f b (i-1) (b*y)
325 _ ^ _ = error "Prelude.^: negative exponent"
327 {-# SPECIALISE (^^) ::
328 Rational -> Int -> Rational #-}
329 (^^) :: (Fractional a, Integral b) => a -> b -> a
330 x ^^ n = if n >= 0 then x^n else recip (x^(negate n))
333 -------------------------------------------------------
334 gcd :: (Integral a) => a -> a -> a
335 gcd 0 0 = error "Prelude.gcd: gcd 0 0 is undefined"
336 gcd x y = gcd' (abs x) (abs y)
338 gcd' a b = gcd' b (a `rem` b)
340 lcm :: (Integral a) => a -> a -> a
341 {-# SPECIALISE lcm :: Int -> Int -> Int #-}
344 lcm x y = abs ((x `quot` (gcd x y)) * y)
348 "gcd/Int->Int->Int" gcd = gcdInt
349 "gcd/Integer->Integer->Integer" gcd = gcdInteger
350 "lcm/Integer->Integer->Integer" lcm = lcmInteger
353 integralEnumFrom :: (Integral a, Bounded a) => a -> [a]
354 integralEnumFrom n = map fromInteger [toInteger n .. toInteger (maxBound `asTypeOf` n)]
356 integralEnumFromThen :: (Integral a, Bounded a) => a -> a -> [a]
357 integralEnumFromThen n1 n2
358 | i_n2 >= i_n1 = map fromInteger [i_n1, i_n2 .. toInteger (maxBound `asTypeOf` n1)]
359 | otherwise = map fromInteger [i_n1, i_n2 .. toInteger (minBound `asTypeOf` n1)]
364 integralEnumFromTo :: Integral a => a -> a -> [a]
365 integralEnumFromTo n m = map fromInteger [toInteger n .. toInteger m]
367 integralEnumFromThenTo :: Integral a => a -> a -> a -> [a]
368 integralEnumFromThenTo n1 n2 m
369 = map fromInteger [toInteger n1, toInteger n2 .. toInteger m]