\begin{code}
-{-# OPTIONS -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)
--
-----------------------------------------------------------------------------
+-- #hide
module GHC.Real where
-import {-# SOURCE #-} GHC.Err
import GHC.Base
import GHC.Num
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,
- -- and we shouldn't be using defaults anyway
+default () -- Double isn't available yet,
+ -- and we shouldn't be using defaults anyway
\end{code}
%*********************************************************
-%* *
+%* *
\subsection{The @Ratio@ and @Rational@ types}
-%* *
+%* *
%*********************************************************
\begin{code}
-data (Integral a) => Ratio a = !a :% !a deriving (Eq)
-type Rational = Ratio Integer
+-- | Rational numbers, with numerator and denominator of some 'Integral' type.
+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
+-- 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
+(%) :: (Integral a) => a -> a -> Ratio 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}
reduce :: (Integral a) => a -> a -> Ratio a
{-# SPECIALISE reduce :: Integer -> Integer -> Rational #-}
-reduce _ 0 = error "Ratio.%: zero denominator"
-reduce x y = (x `quot` d) :% (y `quot` d)
- where d = gcd x y
+reduce _ 0 = error "Ratio.%: zero denominator"
+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)
+x % y = reduce (x * signum y) (abs y)
-numerator (x :% _) = x
-denominator (_ :% y) = 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
+ -- | 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)
- 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 (q-1, r+d) else qr
- where qr@(q,r) = quotRem n d
-
+ -- | 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
+
+ {-# 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 numbers, supporting real division.
+--
+-- Minimal complete definition: 'fromRational' and ('recip' or @('/')@)
class (Num a) => Fractional a where
- (/) :: a -> a -> a
- recip :: a -> a
- fromRational :: Rational -> a
-
- recip x = 1 / x
- x / y = x * recip y
-
+ -- | 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
+
+ {-# INLINE recip #-}
+ {-# INLINE (/) #-}
+ 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
- properFraction :: (Integral b) => a -> (b,a)
- truncate, round :: (Integral b) => a -> b
- ceiling, 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
-
- 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
+ -- | 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' x@ returns the integer nearest @x@ between zero and @x@
+ truncate :: (Integral b) => a -> b
+ -- | @'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}
These 'numeric' enumerations come straight from the Report
\begin{code}
-numericEnumFrom :: (Fractional a) => a -> [a]
-numericEnumFrom = iterate (+1)
+numericEnumFrom :: (Fractional a) => a -> [a]
+numericEnumFrom n = n `seq` (n : numericEnumFrom (n + 1))
-numericEnumFromThen :: (Fractional a) => a -> a -> [a]
-numericEnumFromThen n m = iterate (+(m-n)) n
+numericEnumFromThen :: (Fractional a) => a -> a -> [a]
+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)
- where
- mid = (e2 - e1) / 2
- pred | e2 > e1 = (<= e3 + mid)
- | otherwise = (>= e3 + mid)
+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 = 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"
-
- x `div` y = x `divInt` y
- x `mod` y = x `modInt` y
-
- a `quotRem` b = a `quotRemInt` b
- a `divMod` b = a `divModInt` b
+ 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 -- Note [Order of tests]
+ -- in GHC.Int
+ | otherwise = a `quotInt` b
+
+ a `rem` b
+ | b == 0 = divZeroError
+ | b == (-1) && a == minBound = overflowError -- Note [Order of tests]
+ -- in GHC.Int
+ | otherwise = a `remInt` b
+
+ a `div` b
+ | b == 0 = divZeroError
+ | b == (-1) && a == minBound = overflowError -- Note [Order of tests]
+ -- in GHC.Int
+ | otherwise = a `divInt` b
+
+ a `mod` b
+ | b == 0 = divZeroError
+ | b == (-1) && a == minBound = overflowError -- Note [Order of tests]
+ -- in GHC.Int
+ | otherwise = a `modInt` b
+
+ a `quotRem` b
+ | b == 0 = divZeroError
+ | b == (-1) && a == minBound = overflowError -- Note [Order of tests]
+ -- in GHC.Int
+ | otherwise = a `quotRemInt` b
+
+ a `divMod` b
+ | b == 0 = divZeroError
+ | b == (-1) && a == minBound = overflowError -- Note [Order of tests]
+ -- in GHC.Int
+ | otherwise = a `divModInt` b
\end{code}
%*********************************************************
-%* *
+%* *
\subsection{Instances for @Integer@}
-%* *
+%* *
%*********************************************************
\begin{code}
instance Real Integer where
- toRational x = x % 1
+ toRational x = x % 1
instance Integral Integer where
- toInteger n = n
+ toInteger n = n
+ _ `quot` 0 = divZeroError
n `quot` d = n `quotInteger` d
+
+ _ `rem` 0 = divZeroError
n `rem` d = n `remInteger` d
- n `div` d = q where (q,_) = divMod n d
- n `mod` d = r where (_,r) = divMod n 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)
- a `divMod` b = a `divModInteger` b
- a `quotRem` b = a `quotRemInteger` b
+ -- use the defaults for div & mod
\end{code}
%*********************************************************
-%* *
+%* *
\subsection{Instances for @Ratio@}
-%* *
+%* *
%*********************************************************
\begin{code}
-instance (Integral a) => Ord (Ratio a) where
+instance (Integral a) => Ord (Ratio a) where
{-# SPECIALIZE instance Ord Rational #-}
- (x:%y) <= (x':%y') = x * y' <= x' * y
- (x:%y) < (x':%y') = x * y' < x' * y
+ (x:%y) <= (x':%y') = x * y' <= x' * y
+ (x:%y) < (x':%y') = x * y' < x' * y
-instance (Integral a) => Num (Ratio a) where
+instance (Integral a) => Num (Ratio a) where
{-# SPECIALIZE instance Num Rational #-}
- (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') = 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
+
+{-# 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
-
-instance (Integral a) => Real (Ratio a) where
+ (x:%y) / (x':%y') = (x*y') % (y*x')
+ 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 #-}
- toRational (x:%y) = toInteger x :% toInteger y
+ toRational (x:%y) = toInteger x :% toInteger y
-instance (Integral a) => RealFrac (Ratio a) where
+instance (Integral a) => RealFrac (Ratio a) where
{-# SPECIALIZE instance RealFrac Rational #-}
properFraction (x:%y) = (fromInteger (toInteger q), r:%y)
- where (q,r) = quotRem x y
+ where (q,r) = quotRem x y
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
-
-instance (Integral a) => Enum (Ratio a) where
+ showsPrec p (x:%y) = showParen (p > ratioPrec) $
+ 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
{-# SPECIALIZE instance Enum Rational #-}
- succ x = x + 1
- pred x = x - 1
+ succ x = x + 1
+ pred x = x - 1
- toEnum n = fromInteger (int2Integer n) :% 1
+ toEnum n = fromIntegral n :% 1
fromEnum = fromInteger . truncate
- enumFrom = numericEnumFrom
- enumFromThen = numericEnumFromThen
- enumFromTo = numericEnumFromTo
- enumFromThenTo = numericEnumFromThenTo
+ enumFrom = numericEnumFrom
+ enumFromThen = numericEnumFromThen
+ enumFromTo = numericEnumFromTo
+ enumFromThenTo = numericEnumFromThenTo
\end{code}
%*********************************************************
-%* *
+%* *
\subsection{Coercions}
-%* *
+%* *
%*********************************************************
\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
\end{code}
%*********************************************************
-%* *
+%* *
\subsection{Overloaded numeric functions}
-%* *
+%* *
%*********************************************************
\begin{code}
-showSigned :: (Real a) => (a -> ShowS) -> Int -> a -> ShowS
-showSigned showPos p x
+-- | 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
-even, odd :: (Integral a) => a -> Bool
-even n = n `rem` 2 == 0
-odd = not . even
+even, odd :: (Integral a) => a -> Bool
+even n = n `rem` 2 == 0
+odd = not . even
-------------------------------------------------------
+-- | raise a number to a non-negative integral power
{-# SPECIALISE (^) ::
- 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"
-
-{-# SPECIALISE (^^) ::
- Rational -> Int -> Rational #-}
-(^^) :: (Fractional a, Integral b) => a -> b -> a
-x ^^ n = if n >= 0 then x^n else recip (x^(negate n))
-
+ 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 = 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
+(^^) :: (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 :: (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)
+-- 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)
-lcm :: (Integral a) => a -> a -> a
+-------------------------------------------------------
+-- | @'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@ = @0@.
+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' 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
-lcm 0 _ = 0
-lcm x y = abs ((x `quot` (gcd x y)) * y)
-
+lcm _ 0 = 0
+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