% ------------------------------------------------------------------------------ % $Id: PrelFloat.lhs,v 1.14 2001/11/20 14:12:48 simonpj Exp$ % % (c) The University of Glasgow, 1994-2000 % \section[PrelNum]{Module @PrelNum@} The types Float Double and the classes Floating RealFloat \begin{code} {-# OPTIONS -fno-implicit-prelude #-} #include "../../includes/ieee-flpt.h" module PrelFloat( module PrelFloat, Float#, Double# ) where import PrelBase import PrelList import PrelEnum import PrelShow import PrelNum import PrelReal import PrelArr import PrelMaybe infixr 8 ** \end{code} %********************************************************* %* * \subsection{Standard numeric classes} %* * %********************************************************* \begin{code} class (Fractional a) => Floating a where pi :: a exp, log, sqrt :: a -> a (**), logBase :: a -> a -> a sin, cos, tan :: a -> a asin, acos, atan :: a -> a sinh, cosh, tanh :: a -> a asinh, acosh, atanh :: a -> a x ** y = exp (log x * y) logBase x y = log y / log x sqrt x = x ** 0.5 tan x = sin x / cos x tanh x = sinh x / cosh x class (RealFrac a, Floating a) => RealFloat a where floatRadix :: a -> Integer floatDigits :: a -> Int floatRange :: a -> (Int,Int) decodeFloat :: a -> (Integer,Int) encodeFloat :: Integer -> Int -> a exponent :: a -> Int significand :: a -> a scaleFloat :: Int -> a -> a isNaN, isInfinite, isDenormalized, isNegativeZero, isIEEE :: a -> Bool atan2 :: a -> a -> a exponent x = if m == 0 then 0 else n + floatDigits x where (m,n) = decodeFloat x significand x = encodeFloat m (negate (floatDigits x)) where (m,_) = decodeFloat x scaleFloat k x = encodeFloat m (n+k) where (m,n) = decodeFloat x atan2 y x | x > 0 = atan (y/x) | x == 0 && y > 0 = pi/2 | x < 0 && y > 0 = pi + atan (y/x) |(x <= 0 && y < 0) || (x < 0 && isNegativeZero y) || (isNegativeZero x && isNegativeZero y) = -atan2 (-y) x | y == 0 && (x < 0 || isNegativeZero x) = pi -- must be after the previous test on zero y | x==0 && y==0 = y -- must be after the other double zero tests | otherwise = x + y -- x or y is a NaN, return a NaN (via +) \end{code} %********************************************************* %* * \subsection{Type @Integer@, @Float@, @Double@} %* * %********************************************************* \begin{code} data Float = F# Float# data Double = D# Double# instance CCallable Float instance CReturnable Float instance CCallable Double instance CReturnable Double \end{code} %********************************************************* %* * \subsection{Type @Float@} %* * %********************************************************* \begin{code} instance Eq Float where (F# x) == (F# y) = x eqFloat# y instance Ord Float where (F# x) compare (F# y) | x ltFloat# y = LT | x eqFloat# y = EQ | otherwise = GT (F# x) < (F# y) = x ltFloat# y (F# x) <= (F# y) = x leFloat# y (F# x) >= (F# y) = x geFloat# y (F# x) > (F# y) = x gtFloat# y instance Num Float where (+) x y = plusFloat x y (-) x y = minusFloat x y negate x = negateFloat x (*) x y = timesFloat x y abs x | x >= 0.0 = x | otherwise = negateFloat x signum x | x == 0.0 = 0 | x > 0.0 = 1 | otherwise = negate 1 {-# INLINE fromInteger #-} fromInteger n = encodeFloat n 0 -- It's important that encodeFloat inlines here, and that -- fromInteger in turn inlines, -- so that if fromInteger is applied to an (S# i) the right thing happens instance Real Float where toRational x = (m%1)*(b%1)^^n where (m,n) = decodeFloat x b = floatRadix x instance Fractional Float where (/) x y = divideFloat x y fromRational x = fromRat x recip x = 1.0 / x {-# RULES "truncate/Float->Int" truncate = float2Int #-} instance RealFrac Float where {-# SPECIALIZE properFraction :: Float -> (Int, Float) #-} {-# SPECIALIZE round :: Float -> Int #-} {-# SPECIALIZE ceiling :: Float -> Int #-} {-# SPECIALIZE floor :: Float -> Int #-} {-# SPECIALIZE properFraction :: Float -> (Integer, Float) #-} {-# SPECIALIZE truncate :: Float -> Integer #-} {-# SPECIALIZE round :: Float -> Integer #-} {-# SPECIALIZE ceiling :: Float -> Integer #-} {-# SPECIALIZE floor :: Float -> Integer #-} properFraction x = case (decodeFloat x) of { (m,n) -> let b = floatRadix x in if n >= 0 then (fromInteger m * fromInteger b ^ n, 0.0) else case (quotRem m (b^(negate n))) of { (w,r) -> (fromInteger w, encodeFloat r n) } } truncate x = case properFraction x of (n,_) -> n round x = case properFraction x of (n,r) -> let m = if r < 0.0 then n - 1 else n + 1 half_down = abs r - 0.5 in case (compare half_down 0.0) of LT -> n EQ -> if even n then n else m GT -> m ceiling x = case properFraction x of (n,r) -> if r > 0.0 then n + 1 else n floor x = case properFraction x of (n,r) -> if r < 0.0 then n - 1 else n instance Floating Float where pi = 3.141592653589793238 exp x = expFloat x log x = logFloat x sqrt x = sqrtFloat x sin x = sinFloat x cos x = cosFloat x tan x = tanFloat x asin x = asinFloat x acos x = acosFloat x atan x = atanFloat x sinh x = sinhFloat x cosh x = coshFloat x tanh x = tanhFloat x (**) x y = powerFloat x y logBase x y = log y / log x asinh x = log (x + sqrt (1.0+x*x)) acosh x = log (x + (x+1.0) * sqrt ((x-1.0)/(x+1.0))) atanh x = log ((x+1.0) / sqrt (1.0-x*x)) instance RealFloat Float where floatRadix _ = FLT_RADIX -- from float.h floatDigits _ = FLT_MANT_DIG -- ditto floatRange _ = (FLT_MIN_EXP, FLT_MAX_EXP) -- ditto decodeFloat (F# f#) = case decodeFloat# f# of (# exp#, s#, d# #) -> (J# s# d#, I# exp#) encodeFloat (S# i) j = int_encodeFloat# i j encodeFloat (J# s# d#) e = encodeFloat# s# d# e exponent x = case decodeFloat x of (m,n) -> if m == 0 then 0 else n + floatDigits x significand x = case decodeFloat x of (m,_) -> encodeFloat m (negate (floatDigits x)) scaleFloat k x = case decodeFloat x of (m,n) -> encodeFloat m (n+k) isNaN x = 0 /= isFloatNaN x isInfinite x = 0 /= isFloatInfinite x isDenormalized x = 0 /= isFloatDenormalized x isNegativeZero x = 0 /= isFloatNegativeZero x isIEEE _ = True instance Show Float where showsPrec x = showSigned showFloat x showList = showList__ (showsPrec 0) \end{code} %********************************************************* %* * \subsection{Type @Double@} %* * %********************************************************* \begin{code} instance Eq Double where (D# x) == (D# y) = x ==## y instance Ord Double where (D# x) compare (D# y) | x <## y = LT | x ==## y = EQ | otherwise = GT (D# x) < (D# y) = x <## y (D# x) <= (D# y) = x <=## y (D# x) >= (D# y) = x >=## y (D# x) > (D# y) = x >## y instance Num Double where (+) x y = plusDouble x y (-) x y = minusDouble x y negate x = negateDouble x (*) x y = timesDouble x y abs x | x >= 0.0 = x | otherwise = negateDouble x signum x | x == 0.0 = 0 | x > 0.0 = 1 | otherwise = negate 1 {-# INLINE fromInteger #-} -- See comments with Num Float fromInteger (S# i#) = case (int2Double# i#) of { d# -> D# d# } fromInteger (J# s# d#) = encodeDouble# s# d# 0 instance Real Double where toRational x = (m%1)*(b%1)^^n where (m,n) = decodeFloat x b = floatRadix x instance Fractional Double where (/) x y = divideDouble x y fromRational x = fromRat x recip x = 1.0 / x instance Floating Double where pi = 3.141592653589793238 exp x = expDouble x log x = logDouble x sqrt x = sqrtDouble x sin x = sinDouble x cos x = cosDouble x tan x = tanDouble x asin x = asinDouble x acos x = acosDouble x atan x = atanDouble x sinh x = sinhDouble x cosh x = coshDouble x tanh x = tanhDouble x (**) x y = powerDouble x y logBase x y = log y / log x asinh x = log (x + sqrt (1.0+x*x)) acosh x = log (x + (x+1.0) * sqrt ((x-1.0)/(x+1.0))) atanh x = log ((x+1.0) / sqrt (1.0-x*x)) {-# RULES "truncate/Double->Int" truncate = double2Int #-} instance RealFrac Double where {-# SPECIALIZE properFraction :: Double -> (Int, Double) #-} {-# SPECIALIZE round :: Double -> Int #-} {-# SPECIALIZE ceiling :: Double -> Int #-} {-# SPECIALIZE floor :: Double -> Int #-} {-# SPECIALIZE properFraction :: Double -> (Integer, Double) #-} {-# SPECIALIZE truncate :: Double -> Integer #-} {-# SPECIALIZE round :: Double -> Integer #-} {-# SPECIALIZE ceiling :: Double -> Integer #-} {-# SPECIALIZE floor :: Double -> Integer #-} properFraction x = case (decodeFloat x) of { (m,n) -> let b = floatRadix x in if n >= 0 then (fromInteger m * fromInteger b ^ n, 0.0) else case (quotRem m (b^(negate n))) of { (w,r) -> (fromInteger w, encodeFloat r n) } } truncate x = case properFraction x of (n,_) -> n round x = case properFraction x of (n,r) -> let m = if r < 0.0 then n - 1 else n + 1 half_down = abs r - 0.5 in case (compare half_down 0.0) of LT -> n EQ -> if even n then n else m GT -> m ceiling x = case properFraction x of (n,r) -> if r > 0.0 then n + 1 else n floor x = case properFraction x of (n,r) -> if r < 0.0 then n - 1 else n instance RealFloat Double where floatRadix _ = FLT_RADIX -- from float.h floatDigits _ = DBL_MANT_DIG -- ditto floatRange _ = (DBL_MIN_EXP, DBL_MAX_EXP) -- ditto decodeFloat (D# x#) = case decodeDouble# x# of (# exp#, s#, d# #) -> (J# s# d#, I# exp#) encodeFloat (S# i) j = int_encodeDouble# i j encodeFloat (J# s# d#) e = encodeDouble# s# d# e exponent x = case decodeFloat x of (m,n) -> if m == 0 then 0 else n + floatDigits x significand x = case decodeFloat x of (m,_) -> encodeFloat m (negate (floatDigits x)) scaleFloat k x = case decodeFloat x of (m,n) -> encodeFloat m (n+k) isNaN x = 0 /= isDoubleNaN x isInfinite x = 0 /= isDoubleInfinite x isDenormalized x = 0 /= isDoubleDenormalized x isNegativeZero x = 0 /= isDoubleNegativeZero x isIEEE _ = True instance Show Double where showsPrec x = showSigned showFloat x showList = showList__ (showsPrec 0) \end{code} %********************************************************* %* * \subsection{@Enum@ instances} %* * %********************************************************* The @Enum@ instances for Floats and Doubles are slightly unusual. The @toEnum@ function truncates numbers to Int. The definitions of @enumFrom@ and @enumFromThen@ allow floats to be used in arithmetic series: [0,0.1 .. 1.0]. However, roundoff errors make these somewhat dubious. This example may have either 10 or 11 elements, depending on how 0.1 is represented. NOTE: The instances for Float and Double do not make use of the default methods for @enumFromTo@ and @enumFromThenTo@, as these rely on there being a non-lossy' conversion to and from Ints. Instead we make use of the 1.2 default methods (back in the days when Enum had Ord as a superclass) for these (@numericEnumFromTo@ and @numericEnumFromThenTo@ below.) \begin{code} instance Enum Float where succ x = x + 1 pred x = x - 1 toEnum = int2Float fromEnum = fromInteger . truncate -- may overflow enumFrom = numericEnumFrom enumFromTo = numericEnumFromTo enumFromThen = numericEnumFromThen enumFromThenTo = numericEnumFromThenTo instance Enum Double where succ x = x + 1 pred x = x - 1 toEnum = int2Double fromEnum = fromInteger . truncate -- may overflow enumFrom = numericEnumFrom enumFromTo = numericEnumFromTo enumFromThen = numericEnumFromThen enumFromThenTo = numericEnumFromThenTo \end{code} %********************************************************* %* * \subsection{Printing floating point} %* * %********************************************************* \begin{code} showFloat :: (RealFloat a) => a -> ShowS showFloat x = showString (formatRealFloat FFGeneric Nothing x) -- These are the format types. This type is not exported. data FFFormat = FFExponent | FFFixed | FFGeneric formatRealFloat :: (RealFloat a) => FFFormat -> Maybe Int -> a -> String formatRealFloat fmt decs x | isNaN x = "NaN" | isInfinite x = if x < 0 then "-Infinity" else "Infinity" | x < 0 || isNegativeZero x = '-':doFmt fmt (floatToDigits (toInteger base) (-x)) | otherwise = doFmt fmt (floatToDigits (toInteger base) x) where base = 10 doFmt format (is, e) = let ds = map intToDigit is in case format of FFGeneric -> doFmt (if e < 0 || e > 7 then FFExponent else FFFixed) (is,e) FFExponent -> case decs of Nothing -> let show_e' = show (e-1) in case ds of "0" -> "0.0e0" [d] -> d : ".0e" ++ show_e' (d:ds') -> d : '.' : ds' ++ "e" ++ show_e' Just dec -> let dec' = max dec 1 in case is of [0] -> '0' :'.' : take dec' (repeat '0') ++ "e0" _ -> let (ei,is') = roundTo base (dec'+1) is (d:ds') = map intToDigit (if ei > 0 then init is' else is') in d:'.':ds' ++ 'e':show (e-1+ei) FFFixed -> let mk0 ls = case ls of { "" -> "0" ; _ -> ls} in case decs of Nothing | e <= 0 -> "0." ++ replicate (-e) '0' ++ ds | otherwise -> let f 0 s rs = mk0 (reverse s) ++ '.':mk0 rs f n s "" = f (n-1) ('0':s) "" f n s (r:rs) = f (n-1) (r:s) rs in f e "" ds Just dec -> let dec' = max dec 0 in if e >= 0 then let (ei,is') = roundTo base (dec' + e) is (ls,rs) = splitAt (e+ei) (map intToDigit is') in mk0 ls ++ (if null rs then "" else '.':rs) else let (ei,is') = roundTo base dec' (replicate (-e) 0 ++ is) d:ds' = map intToDigit (if ei > 0 then is' else 0:is') in d : (if null ds' then "" else '.':ds') roundTo :: Int -> Int -> [Int] -> (Int,[Int]) roundTo base d is = case f d is of x@(0,_) -> x (1,xs) -> (1, 1:xs) where b2 = base div 2 f n [] = (0, replicate n 0) f 0 (x:_) = (if x >= b2 then 1 else 0, []) f n (i:xs) | i' == base = (1,0:ds) | otherwise = (0,i':ds) where (c,ds) = f (n-1) xs i' = c + i -- Based on "Printing Floating-Point Numbers Quickly and Accurately" -- by R.G. Burger and R.K. Dybvig in PLDI 96. -- This version uses a much slower logarithm estimator. It should be improved. -- floatToDigits takes a base and a non-negative RealFloat number, -- and returns a list of digits and an exponent. -- In particular, if x>=0, and -- floatToDigits base x = ([d1,d2,...,dn], e) -- then -- (a) n >= 1 -- (b) x = 0.d1d2...dn * (base**e) -- (c) 0 <= di <= base-1 floatToDigits :: (RealFloat a) => Integer -> a -> ([Int], Int) floatToDigits _ 0 = ([0], 0) floatToDigits base x = let (f0, e0) = decodeFloat x (minExp0, _) = floatRange x p = floatDigits x b = floatRadix x minExp = minExp0 - p -- the real minimum exponent -- Haskell requires that f be adjusted so denormalized numbers -- will have an impossibly low exponent. Adjust for this. (f, e) = let n = minExp - e0 in if n > 0 then (f0 div (b^n), e0+n) else (f0, e0) (r, s, mUp, mDn) = if e >= 0 then let be = b^ e in if f == b^(p-1) then (f*be*b*2, 2*b, be*b, b) else (f*be*2, 2, be, be) else if e > minExp && f == b^(p-1) then (f*b*2, b^(-e+1)*2, b, 1) else (f*2, b^(-e)*2, 1, 1) k = let k0 = if b == 2 && base == 10 then -- logBase 10 2 is slightly bigger than 3/10 so -- the following will err on the low side. Ignoring -- the fraction will make it err even more. -- Haskell promises that p-1 <= logBase b f < p. (p - 1 + e0) * 3 div 10 else ceiling ((log (fromInteger (f+1)) + fromInteger (int2Integer e) * log (fromInteger b)) / log (fromInteger base)) --WAS: fromInt e * log (fromInteger b)) fixup n = if n >= 0 then if r + mUp <= expt base n * s then n else fixup (n+1) else if expt base (-n) * (r + mUp) <= s then n else fixup (n+1) in fixup k0 gen ds rn sN mUpN mDnN = let (dn, rn') = (rn * base) divMod sN mUpN' = mUpN * base mDnN' = mDnN * base in case (rn' < mDnN', rn' + mUpN' > sN) of (True, False) -> dn : ds (False, True) -> dn+1 : ds (True, True) -> if rn' * 2 < sN then dn : ds else dn+1 : ds (False, False) -> gen (dn:ds) rn' sN mUpN' mDnN' rds = if k >= 0 then gen [] r (s * expt base k) mUp mDn else let bk = expt base (-k) in gen [] (r * bk) s (mUp * bk) (mDn * bk) in (map fromIntegral (reverse rds), k) \end{code} %********************************************************* %* * \subsection{Converting from a Rational to a RealFloat %* * %********************************************************* [In response to a request for documentation of how fromRational works, Joe Fasel writes:] A quite reasonable request! This code was added to the Prelude just before the 1.2 release, when Lennart, working with an early version of hbi, noticed that (read . show) was not the identity for floating-point numbers. (There was a one-bit error about half the time.) The original version of the conversion function was in fact simply a floating-point divide, as you suggest above. The new version is, I grant you, somewhat denser. Unfortunately, Joe's code doesn't work! Here's an example: main = putStr (shows (1.82173691287639817263897126389712638972163e-300::Double) "\n") This program prints 0.0000000000000000 instead of 1.8217369128763981e-300 Here's Joe's code: \begin{pseudocode} fromRat :: (RealFloat a) => Rational -> a fromRat x = x' where x' = f e -- If the exponent of the nearest floating-point number to x -- is e, then the significand is the integer nearest xb^(-e), -- where b is the floating-point radix. We start with a good -- guess for e, and if it is correct, the exponent of the -- floating-point number we construct will again be e. If -- not, one more iteration is needed. f e = if e' == e then y else f e' where y = encodeFloat (round (x * (1 % b)^^e)) e (_,e') = decodeFloat y b = floatRadix x' -- We obtain a trial exponent by doing a floating-point -- division of x's numerator by its denominator. The -- result of this division may not itself be the ultimate -- result, because of an accumulation of three rounding -- errors. (s,e) = decodeFloat (fromInteger (numerator x) asTypeOf x' / fromInteger (denominator x)) \end{pseudocode} Now, here's Lennart's code (which works) \begin{code} {-# SPECIALISE fromRat :: Rational -> Double, Rational -> Float #-} fromRat :: (RealFloat a) => Rational -> a fromRat x | x == 0 = encodeFloat 0 0 -- Handle exceptional cases | x < 0 = - fromRat' (-x) -- first. | otherwise = fromRat' x -- Conversion process: -- Scale the rational number by the RealFloat base until -- it lies in the range of the mantissa (as used by decodeFloat/encodeFloat). -- Then round the rational to an Integer and encode it with the exponent -- that we got from the scaling. -- To speed up the scaling process we compute the log2 of the number to get -- a first guess of the exponent. fromRat' :: (RealFloat a) => Rational -> a fromRat' x = r where b = floatRadix r p = floatDigits r (minExp0, _) = floatRange r minExp = minExp0 - p -- the real minimum exponent xMin = toRational (expt b (p-1)) xMax = toRational (expt b p) p0 = (integerLogBase b (numerator x) - integerLogBase b (denominator x) - p) max minExp f = if p0 < 0 then 1 % expt b (-p0) else expt b p0 % 1 (x', p') = scaleRat (toRational b) minExp xMin xMax p0 (x / f) r = encodeFloat (round x') p' -- Scale x until xMin <= x < xMax, or p (the exponent) <= minExp. scaleRat :: Rational -> Int -> Rational -> Rational -> Int -> Rational -> (Rational, Int) scaleRat b minExp xMin xMax p x | p <= minExp = (x, p) | x >= xMax = scaleRat b minExp xMin xMax (p+1) (x/b) | x < xMin = scaleRat b minExp xMin xMax (p-1) (x*b) | otherwise = (x, p) -- Exponentiation with a cache for the most common numbers. minExpt, maxExpt :: Int minExpt = 0 maxExpt = 1100 expt :: Integer -> Int -> Integer expt base n = if base == 2 && n >= minExpt && n <= maxExpt then expts!n else base^n expts :: Array Int Integer expts = array (minExpt,maxExpt) [(n,2^n) | n <- [minExpt .. maxExpt]] -- Compute the (floor of the) log of i in base b. -- Simplest way would be just divide i by b until it's smaller then b, but that would -- be very slow! We are just slightly more clever. integerLogBase :: Integer -> Integer -> Int integerLogBase b i | i < b = 0 | otherwise = doDiv (i div (b^l)) l where -- Try squaring the base first to cut down the number of divisions. l = 2 * integerLogBase (b*b) i doDiv :: Integer -> Int -> Int doDiv x y | x < b = y | otherwise = doDiv (x div` b) (y+1) \end{code} %********************************************************* %* * \subsection{Floating point numeric primops} %* * %********************************************************* Definitions of the boxed PrimOps; these will be used in the case of partial applications, etc. \begin{code} plusFloat, minusFloat, timesFloat, divideFloat :: Float -> Float -> Float plusFloat (F# x) (F# y) = F# (plusFloat# x y) minusFloat (F# x) (F# y) = F# (minusFloat# x y) timesFloat (F# x) (F# y) = F# (timesFloat# x y) divideFloat (F# x) (F# y) = F# (divideFloat# x y) {-# RULES "plusFloat x 0.0" forall x#. plusFloat# x# 0.0# = x# "plusFloat 0.0 x" forall x#. plusFloat# 0.0# x# = x# "minusFloat x 0.0" forall x#. minusFloat# x# 0.0# = x# "minusFloat x x" forall x#. minusFloat# x# x# = 0.0# "timesFloat x 0.0" forall x#. timesFloat# x# 0.0# = 0.0# "timesFloat0.0 x" forall x#. timesFloat# 0.0# x# = 0.0# "timesFloat x 1.0" forall x#. timesFloat# x# 1.0# = x# "timesFloat 1.0 x" forall x#. timesFloat# 1.0# x# = x# "divideFloat x 1.0" forall x#. divideFloat# x# 1.0# = x# #-} negateFloat :: Float -> Float negateFloat (F# x) = F# (negateFloat# x) gtFloat, geFloat, eqFloat, neFloat, ltFloat, leFloat :: Float -> Float -> Bool gtFloat (F# x) (F# y) = gtFloat# x y geFloat (F# x) (F# y) = geFloat# x y eqFloat (F# x) (F# y) = eqFloat# x y neFloat (F# x) (F# y) = neFloat# x y ltFloat (F# x) (F# y) = ltFloat# x y leFloat (F# x) (F# y) = leFloat# x y float2Int :: Float -> Int float2Int (F# x) = I# (float2Int# x) int2Float :: Int -> Float int2Float (I# x) = F# (int2Float# x) expFloat, logFloat, sqrtFloat :: Float -> Float sinFloat, cosFloat, tanFloat :: Float -> Float asinFloat, acosFloat, atanFloat :: Float -> Float sinhFloat, coshFloat, tanhFloat :: Float -> Float expFloat (F# x) = F# (expFloat# x) logFloat (F# x) = F# (logFloat# x) sqrtFloat (F# x) = F# (sqrtFloat# x) sinFloat (F# x) = F# (sinFloat# x) cosFloat (F# x) = F# (cosFloat# x) tanFloat (F# x) = F# (tanFloat# x) asinFloat (F# x) = F# (asinFloat# x) acosFloat (F# x) = F# (acosFloat# x) atanFloat (F# x) = F# (atanFloat# x) sinhFloat (F# x) = F# (sinhFloat# x) coshFloat (F# x) = F# (coshFloat# x) tanhFloat (F# x) = F# (tanhFloat# x) powerFloat :: Float -> Float -> Float powerFloat (F# x) (F# y) = F# (powerFloat# x y) -- definitions of the boxed PrimOps; these will be -- used in the case of partial applications, etc. plusDouble, minusDouble, timesDouble, divideDouble :: Double -> Double -> Double plusDouble (D# x) (D# y) = D# (x +## y) minusDouble (D# x) (D# y) = D# (x -## y) timesDouble (D# x) (D# y) = D# (x *## y) divideDouble (D# x) (D# y) = D# (x /## y) {-# RULES "plusDouble x 0.0" forall x#. (+##) x# 0.0## = x# "plusDouble 0.0 x" forall x#. (+##) 0.0## x# = x# "minusDouble x 0.0" forall x#. (-##) x# 0.0## = x# "minusDouble x x" forall x#. (-##) x# x# = 0.0## "timesDouble x 0.0" forall x#. (*##) x# 0.0## = 0.0## "timesDouble 0.0 x" forall x#. (*##) 0.0## x# = 0.0## "timesDouble x 1.0" forall x#. (*##) x# 1.0## = x# "timesDouble 1.0 x" forall x#. (*##) 1.0## x# = x# "divideDouble x 1.0" forall x#. (/##) x# 1.0## = x# #-} negateDouble :: Double -> Double negateDouble (D# x) = D# (negateDouble# x) gtDouble, geDouble, eqDouble, neDouble, leDouble, ltDouble :: Double -> Double -> Bool gtDouble (D# x) (D# y) = x >## y geDouble (D# x) (D# y) = x >=## y eqDouble (D# x) (D# y) = x ==## y neDouble (D# x) (D# y) = x /=## y ltDouble (D# x) (D# y) = x <## y leDouble (D# x) (D# y) = x <=## y double2Int :: Double -> Int double2Int (D# x) = I# (double2Int# x) int2Double :: Int -> Double int2Double (I# x) = D# (int2Double# x) double2Float :: Double -> Float double2Float (D# x) = F# (double2Float# x) float2Double :: Float -> Double float2Double (F# x) = D# (float2Double# x) expDouble, logDouble, sqrtDouble :: Double -> Double sinDouble, cosDouble, tanDouble :: Double -> Double asinDouble, acosDouble, atanDouble :: Double -> Double sinhDouble, coshDouble, tanhDouble :: Double -> Double expDouble (D# x) = D# (expDouble# x) logDouble (D# x) = D# (logDouble# x) sqrtDouble (D# x) = D# (sqrtDouble# x) sinDouble (D# x) = D# (sinDouble# x) cosDouble (D# x) = D# (cosDouble# x) tanDouble (D# x) = D# (tanDouble# x) asinDouble (D# x) = D# (asinDouble# x) acosDouble (D# x) = D# (acosDouble# x) atanDouble (D# x) = D# (atanDouble# x) sinhDouble (D# x) = D# (sinhDouble# x) coshDouble (D# x) = D# (coshDouble# x) tanhDouble (D# x) = D# (tanhDouble# x) powerDouble :: Double -> Double -> Double powerDouble (D# x) (D# y) = D# (x **## y) \end{code} \begin{code} foreign import ccall "__encodeFloat" unsafe encodeFloat# :: Int# -> ByteArray# -> Int -> Float foreign import ccall "__int_encodeFloat" unsafe int_encodeFloat# :: Int# -> Int -> Float foreign import ccall "isFloatNaN" unsafe isFloatNaN :: Float -> Int foreign import ccall "isFloatInfinite" unsafe isFloatInfinite :: Float -> Int foreign import ccall "isFloatDenormalized" unsafe isFloatDenormalized :: Float -> Int foreign import ccall "isFloatNegativeZero" unsafe isFloatNegativeZero :: Float -> Int foreign import ccall "__encodeDouble" unsafe encodeDouble# :: Int# -> ByteArray# -> Int -> Double foreign import ccall "__int_encodeDouble" unsafe int_encodeDouble# :: Int# -> Int -> Double foreign import ccall "isDoubleNaN" unsafe isDoubleNaN :: Double -> Int foreign import ccall "isDoubleInfinite" unsafe isDoubleInfinite :: Double -> Int foreign import ccall "isDoubleDenormalized" unsafe isDoubleDenormalized :: Double -> Int foreign import ccall "isDoubleNegativeZero" unsafe isDoubleNegativeZero :: Double -> Int \end{code} %********************************************************* %* * \subsection{Coercion rules} %* * %********************************************************* \begin{code} {-# RULES "fromIntegral/Int->Float" fromIntegral = int2Float "fromIntegral/Int->Double" fromIntegral = int2Double "realToFrac/Float->Float" realToFrac = id :: Float -> Float "realToFrac/Float->Double" realToFrac = float2Double "realToFrac/Double->Float" realToFrac = double2Float "realToFrac/Double->Double" realToFrac = id :: Double -> Double #-} \end{code}