\begin{code}
-{-# OPTIONS_GHC -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)
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,
+default () -- Double isn't available yet,
-- and we shouldn't be using defaults anyway
\end{code}
\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
infinity = 1 :% 0
notANumber = 0 :% 0
--- Use :%, not % for Inf/NaN; the latter would
--- immediately lead to a runtime error, because it normalises.
+-- Use :%, not % for Inf/NaN; the latter would
+-- immediately lead to a runtime error, because it normalises.
\end{code}
-- | 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' a) => a@.
fromRational :: Rational -> a
+ {-# INLINE recip #-}
+ {-# INLINE (/) #-}
recip x = 1 / x
x / y = x * recip y
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@
+ -- | @'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}
\begin{code}
numericEnumFrom :: (Fractional a) => a -> [a]
-numericEnumFrom = iterate (+1)
+numericEnumFrom n = n `seq` (n : numericEnumFrom (n + 1))
numericEnumFromThen :: (Fractional a) => a -> a -> [a]
-numericEnumFromThen n m = iterate (+(m-n)) n
+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)
+numericEnumFromThenTo e1 e2 e3
+ = takeWhile predicate (numericEnumFromThen e1 e2)
where
mid = (e2 - e1) / 2
- pred | e2 >= e1 = (<= e3 + mid)
- | otherwise = (>= e3 + mid)
+ predicate | e2 >= e1 = (<= e3 + mid)
+ | otherwise = (>= e3 + mid)
\end{code}
toRational x = toInteger x % 1
instance Integral Int where
- toInteger i = int2Integer i -- give back a full-blown Integer
+ toInteger (I# i) = smallInteger i
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}
instance Integral Integer where
toInteger n = n
- a `quot` 0 = divZeroError
+ _ `quot` 0 = divZeroError
n `quot` d = n `quotInteger` d
- a `rem` 0 = divZeroError
+ _ `rem` 0 = divZeroError
n `rem` d = n `remInteger` d
- a `divMod` 0 = divZeroError
- a `divMod` b = a `divModInteger` b
+ _ `divMod` 0 = divZeroError
+ a `divMod` b = case a `divModInteger` b of
+ (# x, y #) -> (x, y)
- a `quotRem` 0 = divZeroError
- a `quotRem` b = a `quotRemInteger` b
+ _ `quotRem` 0 = divZeroError
+ a `quotRem` b = case a `quotRemInteger` b of
+ (# q, r #) -> (q, r)
-- use the defaults for div & mod
\end{code}
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
+ 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 #-}
instance (Integral a) => Show (Ratio a) where
{-# SPECIALIZE instance Show Rational #-}
showsPrec p (x:%y) = showParen (p > ratioPrec) $
- showsPrec ratioPrec1 x .
- showString "%" . -- H98 report has spaces round the %
- -- but we removed them [May 04]
+ 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
succ x = x + 1
pred x = x - 1
- toEnum n = fromInteger (int2Integer n) :% 1
+ toEnum n = fromIntegral n :% 1
fromEnum = fromInteger . truncate
enumFrom = numericEnumFrom
-> Int -- ^ the precedence of the enclosing context
-> a -- ^ the value to show
-> ShowS
-showSigned showPos p x
+showSigned showPos p x
| x < 0 = showParen (p > 6) (showChar '-' . showPos (-x))
| otherwise = showPos x
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"
+{-# 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
-{-# 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. -}
-------------------------------------------------------
--- | @'gcd' x y@ is the greatest (positive) integer that divides both @x@
+-- 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 (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@ raises a runtime error.
+-- @'gcd' 0 4@ = @4@. @'gcd' 0 0@ = @0@.
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)
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)]
projects / ghc-base.git / blobdiff