2 {-# LANGUAGE CPP, NoImplicitPrelude, MagicHash, UnboxedTuples #-}
3 {-# OPTIONS_HADDOCK hide #-}
4 -----------------------------------------------------------------------------
7 -- Copyright : (c) The University of Glasgow, 1994-2002
8 -- License : see libraries/base/LICENSE
10 -- Maintainer : cvs-ghc@haskell.org
11 -- Stability : internal
12 -- Portability : non-portable (GHC Extensions)
14 -- The types 'Ratio' and 'Rational', and the classes 'Real', 'Fractional',
15 -- 'Integral', and 'RealFrac'.
17 -----------------------------------------------------------------------------
30 infixl 7 /, `quot`, `rem`, `div`, `mod`
33 default () -- Double isn't available yet,
34 -- and we shouldn't be using defaults anyway
38 %*********************************************************
40 \subsection{The @Ratio@ and @Rational@ types}
42 %*********************************************************
45 -- | Rational numbers, with numerator and denominator of some 'Integral' type.
46 data (Integral a) => Ratio a = !a :% !a deriving (Eq)
48 -- | Arbitrary-precision rational numbers, represented as a ratio of
49 -- two 'Integer' values. A rational number may be constructed using
51 type Rational = Ratio Integer
53 ratioPrec, ratioPrec1 :: Int
54 ratioPrec = 7 -- Precedence of ':%' constructor
55 ratioPrec1 = ratioPrec + 1
57 infinity, notANumber :: Rational
61 -- Use :%, not % for Inf/NaN; the latter would
62 -- immediately lead to a runtime error, because it normalises.
67 -- | Forms the ratio of two integral numbers.
68 {-# SPECIALISE (%) :: Integer -> Integer -> Rational #-}
69 (%) :: (Integral a) => a -> a -> Ratio a
71 -- | Extract the numerator of the ratio in reduced form:
72 -- the numerator and denominator have no common factor and the denominator
74 numerator :: (Integral a) => Ratio a -> a
76 -- | Extract the denominator of the ratio in reduced form:
77 -- the numerator and denominator have no common factor and the denominator
79 denominator :: (Integral a) => Ratio a -> a
82 \tr{reduce} is a subsidiary function used only in this module .
83 It normalises a ratio by dividing both numerator and denominator by
84 their greatest common divisor.
87 reduce :: (Integral a) => a -> a -> Ratio a
88 {-# SPECIALISE reduce :: Integer -> Integer -> Rational #-}
89 reduce _ 0 = error "Ratio.%: zero denominator"
90 reduce x y = (x `quot` d) :% (y `quot` d)
95 x % y = reduce (x * signum y) (abs y)
97 numerator (x :% _) = x
98 denominator (_ :% y) = y
102 %*********************************************************
104 \subsection{Standard numeric classes}
106 %*********************************************************
109 class (Num a, Ord a) => Real a where
110 -- | the rational equivalent of its real argument with full precision
111 toRational :: a -> Rational
113 -- | Integral numbers, supporting integer division.
115 -- Minimal complete definition: 'quotRem' and 'toInteger'
116 class (Real a, Enum a) => Integral a where
117 -- | integer division truncated toward zero
119 -- | integer remainder, satisfying
121 -- > (x `quot` y)*y + (x `rem` y) == x
123 -- | integer division truncated toward negative infinity
125 -- | integer modulus, satisfying
127 -- > (x `div` y)*y + (x `mod` y) == x
129 -- | simultaneous 'quot' and 'rem'
130 quotRem :: a -> a -> (a,a)
131 -- | simultaneous 'div' and 'mod'
132 divMod :: a -> a -> (a,a)
133 -- | conversion to 'Integer'
134 toInteger :: a -> Integer
140 n `quot` d = q where (q,_) = quotRem n d
141 n `rem` d = r where (_,r) = quotRem n d
142 n `div` d = q where (q,_) = divMod n d
143 n `mod` d = r where (_,r) = divMod n d
145 divMod n d = if signum r == negate (signum d) then (q-1, r+d) else qr
146 where qr@(q,r) = quotRem n d
148 -- | Fractional numbers, supporting real division.
150 -- Minimal complete definition: 'fromRational' and ('recip' or @('/')@)
151 class (Num a) => Fractional a where
152 -- | fractional division
154 -- | reciprocal fraction
156 -- | Conversion from a 'Rational' (that is @'Ratio' 'Integer'@).
157 -- A floating literal stands for an application of 'fromRational'
158 -- to a value of type 'Rational', so such literals have type
159 -- @('Fractional' a) => a@.
160 fromRational :: Rational -> a
167 -- | Extracting components of fractions.
169 -- Minimal complete definition: 'properFraction'
170 class (Real a, Fractional a) => RealFrac a where
171 -- | The function 'properFraction' takes a real fractional number @x@
172 -- and returns a pair @(n,f)@ such that @x = n+f@, and:
174 -- * @n@ is an integral number with the same sign as @x@; and
176 -- * @f@ is a fraction with the same type and sign as @x@,
177 -- and with absolute value less than @1@.
179 -- The default definitions of the 'ceiling', 'floor', 'truncate'
180 -- and 'round' functions are in terms of 'properFraction'.
181 properFraction :: (Integral b) => a -> (b,a)
182 -- | @'truncate' x@ returns the integer nearest @x@ between zero and @x@
183 truncate :: (Integral b) => a -> b
184 -- | @'round' x@ returns the nearest integer to @x@;
185 -- the even integer if @x@ is equidistant between two integers
186 round :: (Integral b) => a -> b
187 -- | @'ceiling' x@ returns the least integer not less than @x@
188 ceiling :: (Integral b) => a -> b
189 -- | @'floor' x@ returns the greatest integer not greater than @x@
190 floor :: (Integral b) => a -> b
192 {-# INLINE truncate #-}
193 truncate x = m where (m,_) = properFraction x
195 round x = let (n,r) = properFraction x
196 m = if r < 0 then n - 1 else n + 1
197 in case signum (abs r - 0.5) of
199 0 -> if even n then n else m
201 _ -> error "round default defn: Bad value"
203 ceiling x = if r > 0 then n + 1 else n
204 where (n,r) = properFraction x
206 floor x = if r < 0 then n - 1 else n
207 where (n,r) = properFraction x
211 These 'numeric' enumerations come straight from the Report
214 numericEnumFrom :: (Fractional a) => a -> [a]
215 numericEnumFrom n = n `seq` (n : numericEnumFrom (n + 1))
217 numericEnumFromThen :: (Fractional a) => a -> a -> [a]
218 numericEnumFromThen n m = n `seq` m `seq` (n : numericEnumFromThen m (m+m-n))
220 numericEnumFromTo :: (Ord a, Fractional a) => a -> a -> [a]
221 numericEnumFromTo n m = takeWhile (<= m + 1/2) (numericEnumFrom n)
223 numericEnumFromThenTo :: (Ord a, Fractional a) => a -> a -> a -> [a]
224 numericEnumFromThenTo e1 e2 e3
225 = takeWhile predicate (numericEnumFromThen e1 e2)
228 predicate | e2 >= e1 = (<= e3 + mid)
229 | otherwise = (>= e3 + mid)
233 %*********************************************************
235 \subsection{Instances for @Int@}
237 %*********************************************************
240 instance Real Int where
241 toRational x = toInteger x % 1
243 instance Integral Int where
244 toInteger (I# i) = smallInteger i
247 | b == 0 = divZeroError
248 | b == (-1) && a == minBound = overflowError -- Note [Order of tests]
250 | otherwise = a `quotInt` b
253 | b == 0 = divZeroError
254 | b == (-1) && a == minBound = overflowError -- Note [Order of tests]
256 | otherwise = a `remInt` b
259 | b == 0 = divZeroError
260 | b == (-1) && a == minBound = overflowError -- Note [Order of tests]
262 | otherwise = a `divInt` b
265 | b == 0 = divZeroError
266 | b == (-1) && a == minBound = overflowError -- Note [Order of tests]
268 | otherwise = a `modInt` b
271 | b == 0 = divZeroError
272 | b == (-1) && a == minBound = overflowError -- Note [Order of tests]
274 | otherwise = a `quotRemInt` b
277 | b == 0 = divZeroError
278 | b == (-1) && a == minBound = overflowError -- Note [Order of tests]
280 | otherwise = a `divModInt` b
284 %*********************************************************
286 \subsection{Instances for @Integer@}
288 %*********************************************************
291 instance Real Integer where
294 instance Integral Integer where
297 _ `quot` 0 = divZeroError
298 n `quot` d = n `quotInteger` d
300 _ `rem` 0 = divZeroError
301 n `rem` d = n `remInteger` d
303 _ `divMod` 0 = divZeroError
304 a `divMod` b = case a `divModInteger` b of
307 _ `quotRem` 0 = divZeroError
308 a `quotRem` b = case a `quotRemInteger` b of
311 -- use the defaults for div & mod
315 %*********************************************************
317 \subsection{Instances for @Ratio@}
319 %*********************************************************
322 instance (Integral a) => Ord (Ratio a) where
323 {-# SPECIALIZE instance Ord Rational #-}
324 (x:%y) <= (x':%y') = x * y' <= x' * y
325 (x:%y) < (x':%y') = x * y' < x' * y
327 instance (Integral a) => Num (Ratio a) where
328 {-# SPECIALIZE instance Num Rational #-}
329 (x:%y) + (x':%y') = reduce (x*y' + x'*y) (y*y')
330 (x:%y) - (x':%y') = reduce (x*y' - x'*y) (y*y')
331 (x:%y) * (x':%y') = reduce (x * x') (y * y')
332 negate (x:%y) = (-x) :% y
333 abs (x:%y) = abs x :% y
334 signum (x:%_) = signum x :% 1
335 fromInteger x = fromInteger x :% 1
337 {-# RULES "fromRational/id" fromRational = id :: Rational -> Rational #-}
338 instance (Integral a) => Fractional (Ratio a) where
339 {-# SPECIALIZE instance Fractional Rational #-}
340 (x:%y) / (x':%y') = (x*y') % (y*x')
341 recip (0:%_) = error "Ratio.%: zero denominator"
343 | x < 0 = negate y :% negate x
345 fromRational (x:%y) = fromInteger x % fromInteger y
347 instance (Integral a) => Real (Ratio a) where
348 {-# SPECIALIZE instance Real Rational #-}
349 toRational (x:%y) = toInteger x :% toInteger y
351 instance (Integral a) => RealFrac (Ratio a) where
352 {-# SPECIALIZE instance RealFrac Rational #-}
353 properFraction (x:%y) = (fromInteger (toInteger q), r:%y)
354 where (q,r) = quotRem x y
356 instance (Integral a) => Show (Ratio a) where
357 {-# SPECIALIZE instance Show Rational #-}
358 showsPrec p (x:%y) = showParen (p > ratioPrec) $
359 showsPrec ratioPrec1 x .
361 -- H98 report has spaces round the %
362 -- but we removed them [May 04]
363 -- and added them again for consistency with
364 -- Haskell 98 [Sep 08, #1920]
365 showsPrec ratioPrec1 y
367 instance (Integral a) => Enum (Ratio a) where
368 {-# SPECIALIZE instance Enum Rational #-}
372 toEnum n = fromIntegral n :% 1
373 fromEnum = fromInteger . truncate
375 enumFrom = numericEnumFrom
376 enumFromThen = numericEnumFromThen
377 enumFromTo = numericEnumFromTo
378 enumFromThenTo = numericEnumFromThenTo
382 %*********************************************************
384 \subsection{Coercions}
386 %*********************************************************
389 -- | general coercion from integral types
390 fromIntegral :: (Integral a, Num b) => a -> b
391 fromIntegral = fromInteger . toInteger
394 "fromIntegral/Int->Int" fromIntegral = id :: Int -> Int
397 -- | general coercion to fractional types
398 realToFrac :: (Real a, Fractional b) => a -> b
399 realToFrac = fromRational . toRational
402 "realToFrac/Int->Int" realToFrac = id :: Int -> Int
406 %*********************************************************
408 \subsection{Overloaded numeric functions}
410 %*********************************************************
413 -- | Converts a possibly-negative 'Real' value to a string.
414 showSigned :: (Real a)
415 => (a -> ShowS) -- ^ a function that can show unsigned values
416 -> Int -- ^ the precedence of the enclosing context
417 -> a -- ^ the value to show
419 showSigned showPos p x
420 | x < 0 = showParen (p > 6) (showChar '-' . showPos (-x))
421 | otherwise = showPos x
423 even, odd :: (Integral a) => a -> Bool
424 even n = n `rem` 2 == 0
427 -------------------------------------------------------
428 -- | raise a number to a non-negative integral power
429 {-# SPECIALISE (^) ::
430 Integer -> Integer -> Integer,
431 Integer -> Int -> Integer,
432 Int -> Int -> Int #-}
433 {-# INLINABLE (^) #-} -- See Note [Inlining (^)]
434 (^) :: (Num a, Integral b) => a -> b -> a
435 x0 ^ y0 | y0 < 0 = error "Negative exponent"
437 | otherwise = f x0 y0
438 where -- f : x0 ^ y0 = x ^ y
439 f x y | even y = f (x * x) (y `quot` 2)
441 | otherwise = g (x * x) ((y - 1) `quot` 2) x
442 -- g : x0 ^ y0 = (x ^ y) * z
443 g x y z | even y = g (x * x) (y `quot` 2) z
445 | otherwise = g (x * x) ((y - 1) `quot` 2) (x * z)
447 -- | raise a number to an integral power
448 (^^) :: (Fractional a, Integral b) => a -> b -> a
449 {-# INLINABLE (^^) #-} -- See Note [Inlining (^)
450 x ^^ n = if n >= 0 then x^n else recip (x^(negate n))
452 {- Note [Inlining (^)
453 ~~~~~~~~~~~~~~~~~~~~~
454 The INLINABLE pragma allows (^) to be specialised at its call sites.
455 If it is called repeatedly at the same type, that can make a huge
456 difference, because of those constants which can be repeatedly
459 Currently the fromInteger calls are not floated because we get
461 after the gentle round of simplification. -}
463 -------------------------------------------------------
464 -- Special power functions for Rational
469 -- For a legitimate Rational (n :% d), the numerator and denominator are
470 -- coprime, i.e. they have no common prime factor.
471 -- Therefore all powers (n ^ a) and (d ^ b) are also coprime, so it is
472 -- not necessary to compute the greatest common divisor, which would be
473 -- done in the default implementation at each multiplication step.
474 -- Since exponentiation quickly leads to very large numbers and
475 -- calculation of gcds is generally very slow for large numbers,
476 -- avoiding the gcd leads to an order of magnitude speedup relatively
477 -- soon (and an asymptotic improvement overall).
480 -- We cannot use these functions for general Ratio a because that would
481 -- change results in a multitude of cases.
482 -- The cause is that if a and b are coprime, their remainders by any
483 -- positive modulus generally aren't, so in the default implementation
487 -- (17 % 3) ^ 3 :: Ratio Word8
489 -- (17 % 3) ^ 3 = ((17 % 3) ^ 2) * (17 % 3)
490 -- = ((289 `mod` 256) % 9) * (17 % 3)
491 -- = (33 % 9) * (17 % 3)
492 -- = (11 % 3) * (17 % 3)
495 -- ((17^3) `mod` 256) % (3^3) = (4913 `mod` 256) % 27
499 -- Find out whether special-casing for numerator, denominator or
500 -- exponent = 1 (or -1, where that may apply) gains something.
502 -- Special version of (^) for Rational base
503 {-# RULES "(^)/Rational" (^) = (^%^) #-}
504 (^%^) :: Integral a => Rational -> a -> Rational
506 | e < 0 = error "Negative exponent"
508 | otherwise = (n ^ e) :% (d ^ e)
510 -- Special version of (^^) for Rational base
511 {-# RULES "(^^)/Rational" (^^) = (^^%^^) #-}
512 (^^%^^) :: Integral a => Rational -> a -> Rational
514 | e > 0 = (n ^ e) :% (d ^ e)
516 | n > 0 = (d ^ (negate e)) :% (n ^ (negate e))
517 | n == 0 = error "Ratio.%: zero denominator"
518 | otherwise = let nn = d ^ (negate e)
519 dd = (negate n) ^ (negate e)
520 in if even e then (nn :% dd) else (negate nn :% dd)
522 -------------------------------------------------------
523 -- | @'gcd' x y@ is the greatest (positive) integer that divides both @x@
524 -- and @y@; for example @'gcd' (-3) 6@ = @3@, @'gcd' (-3) (-6)@ = @3@,
525 -- @'gcd' 0 4@ = @4@. @'gcd' 0 0@ raises a runtime error.
526 gcd :: (Integral a) => a -> a -> a
527 gcd 0 0 = error "Prelude.gcd: gcd 0 0 is undefined"
528 gcd x y = gcd' (abs x) (abs y)
530 gcd' a b = gcd' b (a `rem` b)
532 -- | @'lcm' x y@ is the smallest positive integer that both @x@ and @y@ divide.
533 lcm :: (Integral a) => a -> a -> a
534 {-# SPECIALISE lcm :: Int -> Int -> Int #-}
537 lcm x y = abs ((x `quot` (gcd x y)) * y)
539 #ifdef OPTIMISE_INTEGER_GCD_LCM
541 "gcd/Int->Int->Int" gcd = gcdInt
542 "gcd/Integer->Integer->Integer" gcd = gcdInteger'
543 "lcm/Integer->Integer->Integer" lcm = lcmInteger
546 gcdInteger' :: Integer -> Integer -> Integer
547 gcdInteger' 0 0 = error "GHC.Real.gcdInteger': gcd 0 0 is undefined"
548 gcdInteger' a b = gcdInteger a b
550 gcdInt :: Int -> Int -> Int
551 gcdInt 0 0 = error "GHC.Real.gcdInt: gcd 0 0 is undefined"
552 gcdInt a b = fromIntegral (gcdInteger (fromIntegral a) (fromIntegral b))
555 integralEnumFrom :: (Integral a, Bounded a) => a -> [a]
556 integralEnumFrom n = map fromInteger [toInteger n .. toInteger (maxBound `asTypeOf` n)]
558 integralEnumFromThen :: (Integral a, Bounded a) => a -> a -> [a]
559 integralEnumFromThen n1 n2
560 | i_n2 >= i_n1 = map fromInteger [i_n1, i_n2 .. toInteger (maxBound `asTypeOf` n1)]
561 | otherwise = map fromInteger [i_n1, i_n2 .. toInteger (minBound `asTypeOf` n1)]
566 integralEnumFromTo :: Integral a => a -> a -> [a]
567 integralEnumFromTo n m = map fromInteger [toInteger n .. toInteger m]
569 integralEnumFromThenTo :: Integral a => a -> a -> a -> [a]
570 integralEnumFromThenTo n1 n2 m
571 = map fromInteger [toInteger n1, toInteger n2 .. toInteger m]