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)
| x < 0 = negate y :% negate x
| otherwise = y :% x
- fromRational (x:%y) = fromInteger x :% fromInteger y
+ fromRational (x:%y) = fromInteger x % fromInteger y
instance (Integral a) => Real (Ratio a) where
{-# SPECIALIZE instance Real Rational #-}
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 = g (x * x) ((y - 1) `quot` 2) (x * z)
-- | raise a number to an integral power
-{-# SPECIALISE (^^) ::
- Rational -> Int -> Rational #-}
(^^) :: (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. -}
+
+-------------------------------------------------------
+-- 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)
-------------------------------------------------------
-- | @'gcd' x y@ is the greatest (positive) integer that divides both @x@
projects / ghc-base.git / blobdiff