\begin{code}
-{-# OPTIONS_GHC -XNoImplicitPrelude #-}
+{-# LANGUAGE CPP, NoImplicitPrelude, MagicHash, UnboxedTuples #-}
{-# OPTIONS_HADDOCK hide #-}
-----------------------------------------------------------------------------
-- |
\begin{code}
-- | Rational numbers, with numerator and denominator of some 'Integral' type.
-data (Integral a) => Ratio a = !a :% !a deriving (Eq)
+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
a `quot` b
| b == 0 = divZeroError
- | a == minBound && b == (-1) = overflowError
+ | b == (-1) && a == minBound = overflowError -- Note [Order of tests]
+ -- in GHC.Int
| otherwise = a `quotInt` b
a `rem` b
| b == 0 = divZeroError
- | a == minBound && b == (-1) = overflowError
+ | b == (-1) && a == minBound = overflowError -- Note [Order of tests]
+ -- in GHC.Int
| otherwise = a `remInt` b
a `div` b
| b == 0 = divZeroError
- | a == minBound && b == (-1) = overflowError
+ | b == (-1) && a == minBound = overflowError -- Note [Order of tests]
+ -- in GHC.Int
| otherwise = a `divInt` b
a `mod` b
| b == 0 = divZeroError
- | a == minBound && b == (-1) = overflowError
+ | b == (-1) && a == minBound = overflowError -- Note [Order of tests]
+ -- in GHC.Int
| otherwise = a `modInt` b
a `quotRem` b
| b == 0 = divZeroError
- | a == minBound && b == (-1) = overflowError
+ | b == (-1) && a == minBound = overflowError -- Note [Order of tests]
+ -- in GHC.Int
| otherwise = a `quotRemInt` b
a `divMod` b
| b == 0 = divZeroError
- | a == minBound && b == (-1) = overflowError
+ | b == (-1) && a == minBound = overflowError -- Note [Order of tests]
+ -- in GHC.Int
| otherwise = a `divModInt` b
\end{code}
in if even e then (nn :% dd) else (negate nn :% dd)
-------------------------------------------------------
--- | @'gcd' x y@ is the greatest (positive) integer that divides both @x@
--- and @y@; for example @'gcd' (-3) 6@ = @3@, @'gcd' (-3) (-6)@ = @3@,
--- @'gcd' 0 4@ = @4@. @'gcd' 0 0@ raises a runtime error.
+-- | @'gcd' x y@ is the non-negative factor of both @x@ and @y@ of which
+-- every common factor of @x@ and @y@ is also a factor; for example
+-- @'gcd' 4 2 = 2@, @'gcd' (-4) 6 = 2@, @'gcd' 0 4@ = @4@. @'gcd' 0 0@ = @0@.
+-- (That is, the common divisor that is \"greatest\" in the divisibility
+-- preordering.)
+--
+-- Note: Since for signed fixed-width integer types, @'abs' 'minBound' < 0@,
+-- the result may be negative if one of the arguments is @'minBound'@ (and
+-- necessarily is if the other is @0@ or @'minBound'@) for such types.
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)
#ifdef OPTIMISE_INTEGER_GCD_LCM
{-# RULES
"gcd/Int->Int->Int" gcd = gcdInt
-"gcd/Integer->Integer->Integer" gcd = gcdInteger'
+"gcd/Integer->Integer->Integer" gcd = gcdInteger
"lcm/Integer->Integer->Integer" lcm = lcmInteger
#-}
-gcdInteger' :: Integer -> Integer -> Integer
-gcdInteger' 0 0 = error "GHC.Real.gcdInteger': gcd 0 0 is undefined"
-gcdInteger' a b = gcdInteger a b
-
gcdInt :: Int -> Int -> Int
-gcdInt 0 0 = error "GHC.Real.gcdInt: gcd 0 0 is undefined"
gcdInt a b = fromIntegral (gcdInteger (fromIntegral a) (fromIntegral b))
#endif