module GHC.Real where
import GHC.Base
import GHC.Num
import GHC.List
import GHC.Enum
import GHC.Show
import GHC.Err
#ifdef OPTIMISE_INTEGER_GCD_LCM
import GHC.Integer.GMP.Internals
#endif
infixr 8 ^, ^^
infixl 7 /, `quot`, `rem`, `div`, `mod`
infixl 7 %
default ()
\end{code} %********************************************************* %* * \subsection{The @Ratio@ and @Rational@ types} %* * %********************************************************* \begin{code}
data Ratio a = !a :% !a deriving (Eq) type Rational = Ratio Integer ratioPrec, ratioPrec1 :: Int ratioPrec = 7 ratioPrec1 = ratioPrec + 1 infinity, notANumber :: Rational infinity = 1 :% 0 notANumber = 0 :% 0\end{code} \begin{code}
(%) :: (Integral a) => a -> a -> Ratio a numerator :: (Integral a) => Ratio a -> a denominator :: (Integral a) => Ratio a -> a\end{code} \tr{reduce} is a subsidiary function used only in this module . It normalises a ratio by dividing both numerator and denominator by their greatest common divisor. \begin{code}
reduce :: (Integral a) => a -> a -> Ratio a
reduce _ 0 = ratioZeroDenominatorError
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) 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
class (Real a, Enum a) => Integral a where
quot :: a -> a -> a
rem :: a -> a -> a
div :: a -> a -> a
mod :: a -> a -> a
quotRem :: a -> a -> (a,a)
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 (q1, r+d) else qr
where qr@(q,r) = quotRem n d
class (Num a) => Fractional a where
(/) :: a -> a -> a
recip :: a -> a
fromRational :: Rational -> a
recip x = 1 / x
x / y = x * recip y
class (Real a, Fractional a) => RealFrac a where
properFraction :: (Integral b) => a -> (b,a)
truncate :: (Integral b) => a -> b
round :: (Integral b) => a -> b
ceiling :: (Integral b) => a -> b
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
_ -> 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 n = n `seq` (n : numericEnumFrom (n + 1))
numericEnumFromThen :: (Fractional a) => a -> a -> [a]
numericEnumFromThen n m = n `seq` m `seq` (n : numericEnumFromThen m (m+mn))
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 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# i) = smallInteger i
a `quot` b
| b == 0 = divZeroError
| b == (1) && a == minBound = overflowError
| otherwise = a `quotInt` b
a `rem` b
| b == 0 = divZeroError
| b == (1) = 0
| otherwise = a `remInt` b
a `div` b
| b == 0 = divZeroError
| b == (1) && a == minBound = overflowError
| otherwise = a `divInt` b
a `mod` b
| b == 0 = divZeroError
| b == (1) = 0
| otherwise = a `modInt` b
a `quotRem` b
| b == 0 = divZeroError
| b == (1) && a == minBound = (overflowError, 0)
| otherwise = a `quotRemInt` b
a `divMod` b
| b == 0 = divZeroError
| b == (1) && a == minBound = (overflowError, 0)
| otherwise = a `divModInt` b
\end{code} %********************************************************* %* * \subsection{Instances for @Word@} %* * %********************************************************* \begin{code}
instance Real Word where
toRational x = toInteger x % 1
instance Integral Word where
quot (W# x#) y@(W# y#)
| y /= 0 = W# (x# `quotWord#` y#)
| otherwise = divZeroError
rem (W# x#) y@(W# y#)
| y /= 0 = W# (x# `remWord#` y#)
| otherwise = divZeroError
div (W# x#) y@(W# y#)
| y /= 0 = W# (x# `quotWord#` y#)
| otherwise = divZeroError
mod (W# x#) y@(W# y#)
| y /= 0 = W# (x# `remWord#` y#)
| otherwise = divZeroError
quotRem (W# x#) y@(W# y#)
| y /= 0 = case x# `quotRemWord#` y# of
(# q, r #) ->
(W# q, W# r)
| otherwise = divZeroError
divMod (W# x#) y@(W# y#)
| y /= 0 = (W# (x# `quotWord#` y#), W# (x# `remWord#` y#))
| otherwise = divZeroError
toInteger (W# x#)
| i# >=# 0# = smallInteger i#
| otherwise = wordToInteger x#
where
!i# = word2Int# x#
instance Enum Word where
succ x
| x /= maxBound = x + 1
| otherwise = succError "Word"
pred x
| x /= minBound = x 1
| otherwise = predError "Word"
toEnum i@(I# i#)
| i >= 0 = W# (int2Word# i#)
| otherwise = toEnumError "Word" i (minBound::Word, maxBound::Word)
fromEnum x@(W# x#)
| x <= fromIntegral (maxBound::Int)
= I# (word2Int# x#)
| otherwise = fromEnumError "Word" x
enumFrom = integralEnumFrom
enumFromThen = integralEnumFromThen
enumFromTo = integralEnumFromTo
enumFromThenTo = integralEnumFromThenTo
\end{code} %********************************************************* %* * \subsection{Instances for @Integer@} %* * %********************************************************* \begin{code}
instance Real Integer where
toRational x = x :% 1
instance Integral Integer where
toInteger n = n
_ `quot` 0 = divZeroError
n `quot` d = n `quotInteger` d
_ `rem` 0 = divZeroError
n `rem` d = n `remInteger` d
_ `div` 0 = divZeroError
n `div` d = n `divInteger` d
_ `mod` 0 = divZeroError
n `mod` d = n `modInteger` 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)
\end{code} %********************************************************* %* * \subsection{Instances for @Ratio@} %* * %********************************************************* \begin{code}
instance (Integral a) => Ord (Ratio a) where
(x:%y) <= (x':%y') = x * y' <= x' * y
(x:%y) < (x':%y') = x * y' < x' * y
instance (Integral a) => Num (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
instance (Integral a) => Fractional (Ratio a) where
(x:%y) / (x':%y') = (x*y') % (y*x')
recip (0:%_) = ratioZeroDenominatorError
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
toRational (x:%y) = toInteger x :% toInteger y
instance (Integral a) => RealFrac (Ratio a) where
properFraction (x:%y) = (fromInteger (toInteger q), r:%y)
where (q,r) = quotRem x y
instance (Integral a, Show a) => Show (Ratio a) where
showsPrec p (x:%y) = showParen (p > ratioPrec) $
showsPrec ratioPrec1 x .
showString " % " .
showsPrec ratioPrec1 y
instance (Integral a) => Enum (Ratio a) where
succ x = x + 1
pred x = x 1
toEnum n = fromIntegral n :% 1
fromEnum = fromInteger . truncate
enumFrom = numericEnumFrom
enumFromThen = numericEnumFromThen
enumFromTo = numericEnumFromTo
enumFromThenTo = numericEnumFromThenTo
\end{code} %********************************************************* %* * \subsection{Coercions} %* * %********************************************************* \begin{code}
fromIntegral :: (Integral a, Num b) => a -> b fromIntegral = fromInteger . toInteger realToFrac :: (Real a, Fractional b) => a -> b realToFrac = fromRational . toRational\end{code} %********************************************************* %* * \subsection{Overloaded numeric functions} %* * %********************************************************* \begin{code}
showSigned :: (Real a)
=> (a -> ShowS)
-> Int
-> a
-> 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
(^) :: (Num a, Integral b) => a -> b -> a
x0 ^ y0 | y0 < 0 = error "Negative exponent"
| y0 == 0 = 1
| otherwise = f x0 y0
where
f x y | even y = f (x * x) (y `quot` 2)
| y == 1 = x
| otherwise = g (x * x) ((y 1) `quot` 2) x
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)
(^^) :: (Fractional a, Integral b) => a -> b -> a
x ^^ n = if n >= 0 then x^n else recip (x^(negate n))
(^%^) :: Integral a => Rational -> a -> Rational
(n :% d) ^%^ e
| e < 0 = error "Negative exponent"
| e == 0 = 1 :% 1
| otherwise = (n ^ e) :% (d ^ e)
(^^%^^) :: 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 = ratioZeroDenominatorError
| otherwise = let nn = d ^ (negate e)
dd = (negate n) ^ (negate e)
in if even e then (nn :% dd) else (negate nn :% dd)
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 :: (Integral a) => a -> a -> a
lcm _ 0 = 0
lcm 0 _ = 0
lcm x y = abs ((x `quot` (gcd x y)) * y)
#ifdef OPTIMISE_INTEGER_GCD_LCM
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)]
integralEnumFromThen :: (Integral a, Bounded a) => a -> a -> [a]
integralEnumFromThen n1 n2
| i_n2 >= i_n1 = map fromInteger [i_n1, i_n2 .. toInteger (maxBound `asTypeOf` n1)]
| otherwise = map fromInteger [i_n1, i_n2 .. toInteger (minBound `asTypeOf` n1)]
where
i_n1 = toInteger n1
i_n2 = toInteger n2
integralEnumFromTo :: Integral a => a -> a -> [a]
integralEnumFromTo n m = map fromInteger [toInteger n .. toInteger m]
integralEnumFromThenTo :: Integral a => a -> a -> a -> [a]
integralEnumFromThenTo n1 n2 m
= map fromInteger [toInteger n1, toInteger n2 .. toInteger m]
\end{code}
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