X-Git-Url: http://git.megacz.com/?a=blobdiff_plain;f=GHC%2FReal.lhs;h=51d7db404cab4e09ca432662484dcafe941d47f6;hb=7a97ec4b12e1fbec5505f82032cf4dc435b5a60c;hp=13be1422367879db28fe0135cb50a8e98f142755;hpb=b706340c451952adf230b5b8daecad8a1f34d714;p=ghc-base.git diff --git a/GHC/Real.lhs b/GHC/Real.lhs index 13be142..51d7db4 100644 --- a/GHC/Real.lhs +++ b/GHC/Real.lhs @@ -1,11 +1,12 @@ \begin{code} -{-# OPTIONS -fno-implicit-prelude #-} +{-# LANGUAGE CPP, NoImplicitPrelude, MagicHash, UnboxedTuples #-} +{-# OPTIONS_HADDOCK hide #-} ----------------------------------------------------------------------------- -- | -- Module : GHC.Real --- Copyright : (c) The FFI Task Force, 1994-2002 +-- Copyright : (c) The University of Glasgow, 1994-2002 -- License : see libraries/base/LICENSE --- +-- -- Maintainer : cvs-ghc@haskell.org -- Stability : internal -- Portability : non-portable (GHC Extensions) @@ -15,39 +16,67 @@ -- ----------------------------------------------------------------------------- +-- #hide module GHC.Real where -import {-# SOURCE #-} GHC.Err import GHC.Base import GHC.Num import GHC.List import GHC.Enum import GHC.Show +import GHC.Err infixr 8 ^, ^^ infixl 7 /, `quot`, `rem`, `div`, `mod` +infixl 7 % -default () -- Double isn't available yet, - -- and we shouldn't be using defaults anyway +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 +-- | Rational numbers, with numerator and denominator of some 'Integral' type. +data 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 -numerator, denominator :: (Integral a) => Ratio a -> a +(%) :: (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 . @@ -57,212 +86,307 @@ 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 +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) +x % y = reduce (x * signum y) (abs y) -numerator (x :% _) = x -denominator (_ :% y) = 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 + -- | 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 - 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 - + -- | 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 + + {-# INLINE quot #-} + {-# INLINE rem #-} + {-# INLINE div #-} + {-# INLINE mod #-} + 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 - (/) :: a -> a -> a - recip :: a -> a - fromRational :: Rational -> a - - recip x = 1 / x - x / y = x * recip y - + -- | 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 + + {-# INLINE recip #-} + {-# INLINE (/) #-} + 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 - 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 + -- | 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@; + -- the even integer if @x@ is equidistant between two integers + 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 + + {-# INLINE truncate #-} + 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 + _ -> error "round default defn: Bad value" + + 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) +numericEnumFrom :: (Fractional a) => a -> [a] +numericEnumFrom n = n `seq` (n : numericEnumFrom (n + 1)) -numericEnumFromThen :: (Fractional a) => a -> a -> [a] -numericEnumFromThen n m = iterate (+(m-n)) n +numericEnumFromThen :: (Fractional a) => a -> a -> [a] +numericEnumFromThen n m = n `seq` m `seq` (n : numericEnumFromThen m (m+m-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) +numericEnumFromThenTo e1 e2 e3 + = takeWhile predicate (numericEnumFromThen e1 e2) + where + mid = (e2 - e1) / 2 + predicate | 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 + toRational x = toInteger x % 1 + +instance Integral Int where + toInteger (I# i) = smallInteger i + + a `quot` b + | b == 0 = divZeroError + | b == (-1) && a == minBound = overflowError -- Note [Order of tests] + -- in GHC.Int + | otherwise = a `quotInt` b + + a `rem` b + | b == 0 = divZeroError + | b == (-1) && a == minBound = overflowError -- Note [Order of tests] + -- in GHC.Int + | otherwise = a `remInt` b + + a `div` b + | b == 0 = divZeroError + | b == (-1) && a == minBound = overflowError -- Note [Order of tests] + -- in GHC.Int + | otherwise = a `divInt` b + + a `mod` b + | b == 0 = divZeroError + | b == (-1) && a == minBound = overflowError -- Note [Order of tests] + -- in GHC.Int + | otherwise = a `modInt` b + + a `quotRem` b + | b == 0 = divZeroError + | b == (-1) && a == minBound = overflowError -- Note [Order of tests] + -- in GHC.Int + | otherwise = a `quotRemInt` b + + a `divMod` b + | b == 0 = divZeroError + | b == (-1) && a == minBound = overflowError -- Note [Order of tests] + -- in GHC.Int + | otherwise = a `divModInt` b \end{code} %********************************************************* -%* * +%* * \subsection{Instances for @Integer@} -%* * +%* * %********************************************************* \begin{code} instance Real Integer where - toRational x = x % 1 + toRational x = x % 1 instance Integral Integer where - toInteger n = n + toInteger n = n + _ `quot` 0 = divZeroError n `quot` d = n `quotInteger` d + + _ `rem` 0 = divZeroError n `rem` d = n `remInteger` d - n `div` d = q where (q,_) = divMod n d - n `mod` d = r where (_,r) = divMod n d + _ `divMod` 0 = divZeroError + a `divMod` b = case a `divModInteger` b of + (# x, y #) -> (x, y) + + _ `quotRem` 0 = divZeroError + a `quotRem` b = case a `quotRemInteger` b of + (# q, r #) -> (q, r) - a `divMod` b = a `divModInteger` b - 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 +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 + (x:%y) <= (x':%y') = x * y' <= x' * y + (x:%y) < (x':%y') = x * y' < x' * y -instance (Integral a) => Num (Ratio a) where +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 + (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 + +{-# RULES "fromRational/id" fromRational = id :: Rational -> Rational #-} +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 + (x:%y) / (x':%y') = (x*y') % (y*x') + recip (0:%_) = error "Ratio.%: zero denominator" + recip (x:%y) + | x < 0 = negate y :% negate x + | otherwise = 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 + toRational (x:%y) = toInteger x :% toInteger y -instance (Integral a) => RealFrac (Ratio a) where +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 + 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 + showsPrec p (x:%y) = showParen (p > ratioPrec) $ + showsPrec ratioPrec1 x . + showString " % " . + -- H98 report has spaces round the % + -- but we removed them [May 04] + -- and added them again for consistency with + -- Haskell 98 [Sep 08, #1920] + showsPrec ratioPrec1 y + +instance (Integral a) => Enum (Ratio a) where {-# SPECIALIZE instance Enum Rational #-} - succ x = x + 1 - pred x = x - 1 + succ x = x + 1 + pred x = x - 1 - toEnum n = fromInteger (int2Integer n) :% 1 + toEnum n = fromIntegral n :% 1 fromEnum = fromInteger . truncate - enumFrom = numericEnumFrom - enumFromThen = numericEnumFromThen - enumFromTo = numericEnumFromTo - enumFromThenTo = numericEnumFromThenTo + 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 @@ -270,6 +394,7 @@ fromIntegral = fromInteger . toInteger "fromIntegral/Int->Int" fromIntegral = id :: Int -> Int #-} +-- | general coercion to fractional types realToFrac :: (Real a, Fractional b) => a -> b realToFrac = fromRational . toRational @@ -279,61 +404,148 @@ realToFrac = fromRational . toRational \end{code} %********************************************************* -%* * +%* * \subsection{Overloaded numeric functions} -%* * +%* * %********************************************************* \begin{code} -showSigned :: (Real a) => (a -> ShowS) -> Int -> a -> ShowS -showSigned showPos p x +-- | 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 +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" - -{-# SPECIALISE (^^) :: - Rational -> Int -> Rational #-} -(^^) :: (Fractional a, Integral b) => a -> b -> a -x ^^ n = if n >= 0 then x^n else recip (x^(negate n)) - + Integer -> Integer -> Integer, + Integer -> Int -> Integer, + Int -> Int -> Int #-} +{-# INLINABLE (^) #-} -- See Note [Inlining (^)] +(^) :: (Num a, Integral b) => a -> b -> a +x0 ^ y0 | y0 < 0 = error "Negative exponent" + | y0 == 0 = 1 + | otherwise = f x0 y0 + where -- f : x0 ^ y0 = x ^ y + f x y | even y = f (x * x) (y `quot` 2) + | y == 1 = x + | otherwise = g (x * x) ((y - 1) `quot` 2) x + -- g : x0 ^ y0 = (x ^ y) * z + g x y z | even y = g (x * x) (y `quot` 2) z + | y == 1 = x * z + | otherwise = g (x * x) ((y - 1) `quot` 2) (x * z) + +-- | raise a number to an integral power +(^^) :: (Fractional a, Integral b) => a -> b -> a +{-# INLINABLE (^^) #-} -- See Note [Inlining (^) +x ^^ n = if n >= 0 then x^n else recip (x^(negate n)) + +{- Note [Inlining (^) + ~~~~~~~~~~~~~~~~~~~~~ + The INLINABLE pragma allows (^) to be specialised at its call sites. + If it is called repeatedly at the same type, that can make a huge + difference, because of those constants which can be repeatedly + calculated. + + Currently the fromInteger calls are not floated because we get + \d1 d2 x y -> blah + after the gentle round of simplification. -} ------------------------------------------------------- -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) +-- Special power functions for Rational +-- +-- see #4337 +-- +-- Rationale: +-- For a legitimate Rational (n :% d), the numerator and denominator are +-- coprime, i.e. they have no common prime factor. +-- Therefore all powers (n ^ a) and (d ^ b) are also coprime, so it is +-- not necessary to compute the greatest common divisor, which would be +-- done in the default implementation at each multiplication step. +-- Since exponentiation quickly leads to very large numbers and +-- calculation of gcds is generally very slow for large numbers, +-- avoiding the gcd leads to an order of magnitude speedup relatively +-- soon (and an asymptotic improvement overall). +-- +-- Note: +-- We cannot use these functions for general Ratio a because that would +-- change results in a multitude of cases. +-- The cause is that if a and b are coprime, their remainders by any +-- positive modulus generally aren't, so in the default implementation +-- reduction occurs. +-- +-- Example: +-- (17 % 3) ^ 3 :: Ratio Word8 +-- Default: +-- (17 % 3) ^ 3 = ((17 % 3) ^ 2) * (17 % 3) +-- = ((289 `mod` 256) % 9) * (17 % 3) +-- = (33 % 9) * (17 % 3) +-- = (11 % 3) * (17 % 3) +-- = (187 % 9) +-- But: +-- ((17^3) `mod` 256) % (3^3) = (4913 `mod` 256) % 27 +-- = 49 % 27 +-- +-- TODO: +-- Find out whether special-casing for numerator, denominator or +-- exponent = 1 (or -1, where that may apply) gains something. + +-- Special version of (^) for Rational base +{-# RULES "(^)/Rational" (^) = (^%^) #-} +(^%^) :: Integral a => Rational -> a -> Rational +(n :% d) ^%^ e + | e < 0 = error "Negative exponent" + | e == 0 = 1 :% 1 + | otherwise = (n ^ e) :% (d ^ e) + +-- Special version of (^^) for Rational base +{-# RULES "(^^)/Rational" (^^) = (^^%^^) #-} +(^^%^^) :: Integral a => Rational -> a -> Rational +(n :% d) ^^%^^ e + | e > 0 = (n ^ e) :% (d ^ e) + | e == 0 = 1 :% 1 + | n > 0 = (d ^ (negate e)) :% (n ^ (negate e)) + | n == 0 = error "Ratio.%: zero denominator" + | otherwise = let nn = d ^ (negate e) + dd = (negate n) ^ (negate e) + in if even e then (nn :% dd) else (negate nn :% dd) -lcm :: (Integral a) => a -> a -> a +------------------------------------------------------- +-- | @'gcd' x y@ is the greatest (nonnegative) integer that divides both @x@ +-- and @y@; for example @'gcd' (-3) 6@ = @3@, @'gcd' (-3) (-6)@ = @3@, +-- @'gcd' 0 4@ = @4@. @'gcd' 0 0@ = @0@. +gcd :: (Integral a) => a -> a -> a +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) - +lcm _ 0 = 0 +lcm 0 _ = 0 +lcm x y = abs ((x `quot` (gcd x y)) * y) +#ifdef OPTIMISE_INTEGER_GCD_LCM {-# RULES "gcd/Int->Int->Int" gcd = gcdInt "gcd/Integer->Integer->Integer" gcd = gcdInteger "lcm/Integer->Integer->Integer" lcm = lcmInteger #-} +gcdInt :: Int -> Int -> Int +gcdInt a b = fromIntegral (gcdInteger (fromIntegral a) (fromIntegral b)) +#endif + integralEnumFrom :: (Integral a, Bounded a) => a -> [a] integralEnumFrom n = map fromInteger [toInteger n .. toInteger (maxBound `asTypeOf` n)]