From: simonpj Date: Mon, 20 Dec 1999 10:35:47 +0000 (+0000) Subject: [project @ 1999-12-20 10:35:47 by simonpj] X-Git-Tag: Approximately_9120_patches~5370 X-Git-Url: http://git.megacz.com/?a=commitdiff_plain;h=34a4921df0cf2588300ae275ea654ae12395d528;p=ghc-hetmet.git [project @ 1999-12-20 10:35:47 by simonpj] Forgot to remove PrelNumExtra in the last commit --- diff --git a/ghc/lib/std/PrelNumExtra.lhs b/ghc/lib/std/PrelNumExtra.lhs deleted file mode 100644 index c9fa3c5..0000000 --- a/ghc/lib/std/PrelNumExtra.lhs +++ /dev/null @@ -1,998 +0,0 @@ -% -% (c) The AQUA Project, Glasgow University, 1994-1996 -% - -\section[PrelNumExtra]{Module @PrelNumExtra@} - -\begin{code} -{-# OPTIONS -fno-cpr-analyse #-} -{-# OPTIONS -fno-implicit-prelude #-} -{-# OPTIONS -H20m #-} - -#include "../includes/ieee-flpt.h" - -\end{code} - -\begin{code} -module PrelNumExtra where - -import PrelBase -import PrelGHC -import PrelEnum -import PrelShow -import PrelNum -import PrelErr ( error ) -import PrelList -import PrelMaybe -import Maybe ( fromMaybe ) - -import PrelArr ( Array, array, (!) ) -import PrelIOBase ( unsafePerformIO ) -import PrelCCall () -- we need the definitions of CCallable and - -- CReturnable for the foreign calls herein. -\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 - - {-# INLINE fromInt #-} - fromInt i = int2Float i - -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 - -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 RealFrac Float where - - {-# SPECIALIZE properFraction :: Float -> (Int, Float) #-} - {-# SPECIALIZE truncate :: Float -> Int #-} - {-# 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 - -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 - -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 -\end{code} - -%********************************************************* -%* * -\subsection{Type @Double@} -%* * -%********************************************************* - -\begin{code} -instance Show Float where - showsPrec x = showSigned showFloat x - showList = showList__ (showsPrec 0) - -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 n = encodeFloat n 0 - fromInt (I# n#) = case (int2Double# n#) of { d# -> D# d# } - -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)) - -instance RealFrac Double where - - {-# SPECIALIZE properFraction :: Double -> (Int, Double) #-} - {-# SPECIALIZE truncate :: Double -> Int #-} - {-# 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 - -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 - -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{Coercions} -%* * -%********************************************************* - -\begin{code} -{-# SPECIALIZE fromIntegral :: - Int -> Rational, - Integer -> Rational, - Int -> Int, - Int -> Integer, - Int -> Float, - Int -> Double, - Integer -> Int, - Integer -> Integer, - Integer -> Float, - Integer -> Double #-} -fromIntegral :: (Integral a, Num b) => a -> b -fromIntegral = fromInteger . toInteger - -{-# SPECIALIZE realToFrac :: - Double -> Rational, - Rational -> Double, - Float -> Rational, - Rational -> Float, - Rational -> Rational, - Double -> Double, - Double -> Float, - Float -> Float, - Float -> Double #-} -realToFrac :: (Real a, Fractional b) => a -> b -realToFrac = fromRational . toRational -\end{code} - -%********************************************************* -%* * -\subsection{Common code for @Float@ and @Double@} -%* * -%********************************************************* - -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 = fromIntegral - 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 = fromIntegral - fromEnum = fromInteger . truncate -- may overflow - enumFrom = numericEnumFrom - enumFromTo = numericEnumFromTo - enumFromThen = numericEnumFromThen - enumFromThenTo = numericEnumFromThenTo - -numericEnumFrom :: (Fractional a) => a -> [a] -numericEnumFrom = iterate (+1) - -numericEnumFromThen :: (Fractional a) => a -> a -> [a] -numericEnumFromThen n m = iterate (+(m-n)) 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) - -\end{code} - -@approxRational@, applied to two real fractional numbers x and epsilon, -returns the simplest rational number within epsilon of x. A rational -number n%d in reduced form is said to be simpler than another n'%d' if -abs n <= abs n' && d <= d'. Any real interval contains a unique -simplest rational; here, for simplicity, we assume a closed rational -interval. If such an interval includes at least one whole number, then -the simplest rational is the absolutely least whole number. Otherwise, -the bounds are of the form q%1 + r%d and q%1 + r'%d', where abs r < d -and abs r' < d', and the simplest rational is q%1 + the reciprocal of -the simplest rational between d'%r' and d%r. - -\begin{code} -approxRational :: (RealFrac a) => a -> a -> Rational -approxRational rat eps = simplest (rat-eps) (rat+eps) - where simplest x y | y < x = simplest y x - | x == y = xr - | x > 0 = simplest' n d n' d' - | y < 0 = - simplest' (-n') d' (-n) d - | otherwise = 0 :% 1 - where xr = toRational x - n = numerator xr - d = denominator xr - nd' = toRational y - n' = numerator nd' - d' = denominator nd' - - simplest' n d n' d' -- assumes 0 < n%d < n'%d' - | r == 0 = q :% 1 - | q /= q' = (q+1) :% 1 - | otherwise = (q*n''+d'') :% n'' - where (q,r) = quotRem n d - (q',r') = quotRem n' d' - nd'' = simplest' d' r' d r - n'' = numerator nd'' - d'' = denominator nd'' -\end{code} - - -\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) => Real (Ratio a) where - toRational (x:%y) = toInteger x :% toInteger y - -instance (Integral a) => Fractional (Ratio a) where - (x:%y) / (x':%y') = (x*y') % (y*x') - recip (x:%y) = if x < 0 then (-y) :% (-x) else y :% x - fromRational (x:%y) = fromInteger x :% fromInteger y - -instance (Integral a) => RealFrac (Ratio a) where - properFraction (x:%y) = (fromIntegral q, r:%y) - where (q,r) = quotRem x 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 = bounded_iterator True (1) - enumFromThen n m = bounded_iterator (diff >= 0) diff n - where diff = m - n - - -bounded_iterator :: (Ord a, Num a) => Bool -> a -> a -> [a] -bounded_iterator inc step v - | inc && v > new_v = [v] -- oflow - | not inc && v < new_v = [v] -- uflow - | otherwise = v : bounded_iterator inc step new_v - where - new_v = v + step - -ratio_prec :: Int -ratio_prec = 7 - -instance (Integral a) => Show (Ratio a) where - showsPrec p (x:%y) = showParen (p > ratio_prec) - (shows x . showString " % " . shows y) -\end{code} - -@showRational@ converts a Rational to a string that looks like a -floating point number, but without converting to any floating type -(because of the possible overflow). - -From/by Lennart, 94/09/26 - -\begin{code} -showRational :: Int -> Rational -> String -showRational n r = - if r == 0 then - "0.0" - else - let (r', e) = normalize r - in prR n r' e - -startExpExp :: Int -startExpExp = 4 - --- make sure 1 <= r < 10 -normalize :: Rational -> (Rational, Int) -normalize r = if r < 1 then - case norm startExpExp (1 / r) 0 of (r', e) -> (10 / r', -e-1) - else - norm startExpExp r 0 - where norm :: Int -> Rational -> Int -> (Rational, Int) - -- Invariant: x*10^e == original r - norm 0 x e = (x, e) - norm ee x e = - let n = 10^ee - tn = 10^n - in if x >= tn then norm ee (x/tn) (e+n) else norm (ee-1) x e - -prR :: Int -> Rational -> Int -> String -prR n r e | r < 1 = prR n (r*10) (e-1) -- final adjustment -prR n r e | r >= 10 = prR n (r/10) (e+1) -prR n r e0 - | e > 0 && e < 8 = takeN e s ('.' : drop0 (drop e s) []) - | e <= 0 && e > -3 = '0': '.' : takeN (-e) (repeat '0') (drop0 s []) - | otherwise = h : '.' : drop0 t ('e':show e0) - where - s@(h:t) = show ((round (r * 10^n))::Integer) - e = e0+1 - -#ifdef USE_REPORT_PRELUDE - takeN n ls rs = take n ls ++ rs -#else - takeN (I# n#) ls rs = takeUInt_append n# ls rs -#endif - -drop0 :: String -> String -> String -drop0 [] rs = rs -drop0 (c:cs) rs = c : fromMaybe rs (dropTrailing0s cs) --WAS (yuck): reverse (dropWhile (=='0') (reverse cs)) - where - dropTrailing0s [] = Nothing - dropTrailing0s ('0':xs) = - case dropTrailing0s xs of - Nothing -> Nothing - Just ls -> Just ('0':ls) - dropTrailing0s (x:xs) = - case dropTrailing0s xs of - Nothing -> Just [x] - Just ls -> Just (x:ls) - -\end{code} - -[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 - -Lennart's code follows, and it works... - -\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. - -\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{Printing out numbers} -%* * -%********************************************************* - -\begin{code} ---Exported from std library Numeric, defined here to ---avoid mut. rec. between PrelNum and Numeric. -showSigned :: (Real a) => (a -> ShowS) -> Int -> a -> ShowS -showSigned showPos p x - | x < 0 = showParen (p > 6) (showChar '-' . showPos (-x)) - | otherwise = showPos x - -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 -> - 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 : '.' : 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. - --- This function returns a list of digits (Ints in [0..base-1]) and an --- exponent. - -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)) + - fromInt 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 toInt (reverse rds), k) - -\end{code} - -%********************************************************* -%* * -\subsection{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) - -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) - -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}