\begin{code}
-{-# OPTIONS -fno-implicit-prelude #-}
+{-# OPTIONS_GHC -fno-implicit-prelude #-}
-----------------------------------------------------------------------------
-- |
-- Module : GHC.Real
--
-----------------------------------------------------------------------------
+-- #hide
module GHC.Real where
import {-# SOURCE #-} GHC.Err
%*********************************************************
\begin{code}
+-- | Rational numbers, with numerator and denominator of some 'Integral' type.
data (Integral a) => Ratio a = !a :% !a deriving (Eq)
-- | Arbitrary-precision rational numbers, represented as a ratio of
\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
+
+-- | 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 .
\begin{code}
class (Num a, Ord a) => Real a where
+ -- | 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)
+ -- | 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
n `quot` d = q where (q,_) = quotRem 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
+ -- | 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
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
+ -- | 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, round :: (Integral b) => a -> b
- ceiling, floor :: (Integral b) => a -> b
+ -- | @'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 :: (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
truncate x = m where (m,_) = properFraction x
instance Integral Int where
toInteger i = int2Integer i -- give back a full-blown Integer
- a `quot` 0 = divZeroError
- a `quot` b = a `quotInt` b
-
- a `rem` 0 = divZeroError
- a `rem` b = a `remInt` b
-
- a `div` 0 = divZeroError
- a `div` b = a `divInt` b
-
- a `mod` 0 = divZeroError
- a `mod` b = a `modInt` b
-
- a `quotRem` 0 = divZeroError
- a `quotRem` b = a `quotRemInt` b
-
- a `divMod` 0 = divZeroError
- a `divMod` b = a `divModInt` b
+ a `quot` b
+ | b == 0 = divZeroError
+ | a == minBound && b == (-1) = overflowError
+ | otherwise = a `quotInt` b
+
+ a `rem` b
+ | b == 0 = divZeroError
+ | a == minBound && b == (-1) = overflowError
+ | otherwise = a `remInt` b
+
+ a `div` b
+ | b == 0 = divZeroError
+ | a == minBound && b == (-1) = overflowError
+ | otherwise = a `divInt` b
+
+ a `mod` b
+ | b == 0 = divZeroError
+ | a == minBound && b == (-1) = overflowError
+ | otherwise = a `modInt` b
+
+ a `quotRem` b
+ | b == 0 = divZeroError
+ | a == minBound && b == (-1) = overflowError
+ | otherwise = a `quotRemInt` b
+
+ a `divMod` b
+ | b == 0 = divZeroError
+ | a == minBound && b == (-1) = overflowError
+ | otherwise = a `divModInt` b
\end{code}
{-# SPECIALIZE instance Show Rational #-}
showsPrec p (x:%y) = showParen (p > ratioPrec) $
showsPrec ratioPrec1 x .
- showString " % " .
+ showString "%" . -- H98 report has spaces round the %
+ -- but we removed them [May 04]
showsPrec ratioPrec1 y
instance (Integral a) => Enum (Ratio a) where
%*********************************************************
\begin{code}
+-- | general coercion from integral types
fromIntegral :: (Integral a, Num b) => a -> b
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
%*********************************************************
\begin{code}
-showSigned :: (Real a) => (a -> ShowS) -> Int -> a -> ShowS
+-- | 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
odd = not . even
-------------------------------------------------------
+-- | raise a number to a non-negative integral power
{-# SPECIALISE (^) ::
Integer -> Integer -> Integer,
Integer -> Int -> Integer,
| otherwise = f b (i-1) (b*y)
_ ^ _ = error "Prelude.^: negative exponent"
+-- | raise a number to an integral power
{-# SPECIALISE (^^) ::
Rational -> Int -> Rational #-}
(^^) :: (Fractional a, Integral b) => a -> b -> a
-------------------------------------------------------
+-- | @'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 :: (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' 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
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