-% ------------------------------------------------------------------------------
-% $Id: Real.lhs,v 1.3 2002/02/12 15:51:26 simonmar Exp $
-%
-% (c) The University of Glasgow, 1994-2000
-%
-
-\section[GHC.Real]{Module @GHC.Real@}
-
-The types
-
- Ratio, Rational
-
-and the classes
-
- Real
- Integral
- Fractional
- RealFrac
-
-
\begin{code}
-{-# OPTIONS -fno-implicit-prelude #-}
-
+{-# OPTIONS_GHC -fno-implicit-prelude #-}
+-----------------------------------------------------------------------------
+-- |
+-- Module : GHC.Real
+-- Copyright : (c) The FFI Task Force, 1994-2002
+-- License : see libraries/base/LICENSE
+--
+-- Maintainer : cvs-ghc@haskell.org
+-- Stability : internal
+-- Portability : non-portable (GHC Extensions)
+--
+-- The types 'Ratio' and 'Rational', and the classes 'Real', 'Fractional',
+-- 'Integral', and 'RealFrac'.
+--
+-----------------------------------------------------------------------------
+
+-- #hide
module GHC.Real where
import {-# SOURCE #-} GHC.Err
infixr 8 ^, ^^
infixl 7 /, `quot`, `rem`, `div`, `mod`
+infixl 7 %
default () -- Double isn't available yet,
-- and we shouldn't be using defaults anyway
%*********************************************************
\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
+-- two 'Integer' values. A rational number may be constructed using
+-- the '%' operator.
type Rational = Ratio Integer
+
+ratioPrec, ratioPrec1 :: Int
+ratioPrec = 7 -- Precedence of ':%' constructor
+ratioPrec1 = ratioPrec + 1
+
+infinity, notANumber :: Rational
+infinity = 1 :% 0
+notANumber = 0 :% 0
+
+-- Use :%, not % for Inf/NaN; the latter would
+-- immediately lead to a runtime error, because it normalises.
\end{code}
\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
numericEnumFromThenTo e1 e2 e3 = takeWhile pred (numericEnumFromThen e1 e2)
where
mid = (e2 - e1) / 2
- pred | e2 > e1 = (<= e3 + mid)
+ pred | e2 >= e1 = (<= e3 + mid)
| otherwise = (>= e3 + mid)
\end{code}
instance Integral Int where
toInteger i = int2Integer i -- give back a full-blown Integer
- -- Following chks for zero divisor are non-standard (WDP)
- a `quot` b = if b /= 0
- then a `quotInt` b
- else error "Prelude.Integral.quot{Int}: divide by 0"
- a `rem` b = if b /= 0
- then a `remInt` b
- else error "Prelude.Integral.rem{Int}: divide by 0"
+ a `quot` 0 = divZeroError
+ a `quot` b = a `quotInt` b
- x `div` y = x `divInt` y
- x `mod` y = x `modInt` y
+ 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
\end{code}
instance Integral Integer where
toInteger n = n
+ a `quot` 0 = divZeroError
n `quot` d = n `quotInteger` d
- n `rem` d = n `remInteger` d
- n `div` d = q where (q,_) = divMod n d
- n `mod` d = r where (_,r) = divMod n d
+ a `rem` 0 = divZeroError
+ n `rem` d = n `remInteger` d
+ a `divMod` 0 = divZeroError
a `divMod` b = a `divModInteger` b
+
+ a `quotRem` 0 = divZeroError
a `quotRem` b = a `quotRemInteger` b
+
+ -- use the defaults for div & mod
\end{code}
instance (Integral a) => Show (Ratio a) where
{-# SPECIALIZE instance Show Rational #-}
- showsPrec p (x:%y) = showParen (p > ratio_prec)
- (shows x . showString " % " . shows y)
-
-ratio_prec :: Int
-ratio_prec = 7
+ 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 y
instance (Integral a) => Enum (Ratio a) where
{-# SPECIALIZE instance Enum Rational #-}
%*********************************************************
\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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