X-Git-Url: http://git.megacz.com/?a=blobdiff_plain;f=ghc%2Flib%2Fstd%2FPrelNumExtra.lhs;fp=ghc%2Flib%2Fstd%2FPrelNumExtra.lhs;h=20c4b8b0bcf01d505088a4dd8fec1a2b7f07f62c;hb=438596897ebbe25a07e1c82085cfbc5bdb00f09e;hp=0000000000000000000000000000000000000000;hpb=967cc47f37cb93a5e2b6df7822c9a646f0428247;p=ghc-hetmet.git diff --git a/ghc/lib/std/PrelNumExtra.lhs b/ghc/lib/std/PrelNumExtra.lhs new file mode 100644 index 0000000..20c4b8b --- /dev/null +++ b/ghc/lib/std/PrelNumExtra.lhs @@ -0,0 +1,909 @@ +% +% (c) The AQUA Project, Glasgow University, 1994-1996 +% + +\section[PrelNumExtra]{Module @PrelNumExtra@} + +\begin{code} +{-# OPTIONS -fno-implicit-prelude #-} +{-# OPTIONS -H20m #-} + +#include "../includes/ieee-flpt.h" + +\end{code} + +\begin{code} +module PrelNumExtra where + +import PrelBase +import PrelGHC +import PrelNum +import {-# SOURCE #-} PrelErr ( error ) +import PrelList +import PrelMaybe + +import PrelArr ( Array, array, (!) ) +import PrelIOBase ( unsafePerformIO ) +import Ix ( Ix(..) ) +import PrelCCall () -- we need the definitions of CCallable and + -- CReturnable for the _ccall_s 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 + fromInteger n = encodeFloat n 0 + 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 + +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#, a#, s#, d# #) -> (J# a# s# d#, I# exp#) + + encodeFloat (J# a# s# d#) (I# e#) + = case encodeFloat# a# s# d# e# of { flt# -> F# flt# } + + 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::Int) /= unsafePerformIO (_ccall_ isFloatNaN x) {- a _pure_function! -} + isInfinite x = + (0::Int) /= unsafePerformIO (_ccall_ isFloatInfinite x) {- ditto! -} + isDenormalized x = + (0::Int) /= unsafePerformIO (_ccall_ isFloatDenormalized x) -- .. + isNegativeZero x = + (0::Int) /= unsafePerformIO (_ccall_ isFloatNegativeZero x) -- ... + isIEEE x = 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 + 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 #-} + +#if defined(__UNBOXED_INSTANCES__) + {-# SPECIALIZE properFraction :: Double -> (Int#, Double) #-} + {-# SPECIALIZE truncate :: Double -> Int# #-} + {-# SPECIALIZE round :: Double -> Int# #-} + {-# SPECIALIZE ceiling :: Double -> Int# #-} + {-# SPECIALIZE floor :: Double -> Int# #-} +#endif + + 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# d#) + = case decodeDouble# d# of + (# exp#, a#, s#, d# #) -> (J# a# s# d#, I# exp#) + + encodeFloat (J# a# s# d#) (I# e#) + = case encodeDouble# a# s# d# e# of { dbl# -> D# dbl# } + + 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::Int) /= unsafePerformIO (_ccall_ isDoubleNaN x) {- a _pure_function! -} + isInfinite x = + (0::Int) /= unsafePerformIO (_ccall_ isDoubleInfinite x) {- ditto -} + isDenormalized x = + (0::Int) /= unsafePerformIO (_ccall_ isDoubleDenormalized x) -- .. + isNegativeZero x = + (0::Int) /= unsafePerformIO (_ccall_ isDoubleNegativeZero x) -- ... + isIEEE x = 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 fromRealFrac :: + Double -> Rational, + Rational -> Double, + Float -> Rational, + Rational -> Float, + Rational -> Rational, + Double -> Double, + Double -> Float, + Float -> Float, + Float -> Double #-} +fromRealFrac :: (RealFrac a, Fractional b) => a -> b +fromRealFrac = 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 + toEnum = fromIntegral + fromEnum = fromInteger . truncate -- may overflow + enumFrom = numericEnumFrom + enumFromThen = numericEnumFromThen + enumFromThenTo = numericEnumFromThenTo + +instance Enum Double where + toEnum = fromIntegral + fromEnum = fromInteger . truncate -- may overflow + enumFrom = numericEnumFrom + enumFromThen = numericEnumFromThen + enumFromThenTo = numericEnumFromThenTo + +numericEnumFrom :: (Real a) => a -> [a] +numericEnumFromThen :: (Real a) => a -> a -> [a] +numericEnumFromThenTo :: (Real a) => a -> a -> a -> [a] +numericEnumFrom = iterate (+1) +numericEnumFromThen n m = iterate (+(m-n)) n +numericEnumFromThenTo n m p = takeWhile (if m >= n then (<= p) else (>= p)) + (numericEnumFromThen n m) +\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 x eps = simplest (x-eps) (x+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:%y) = 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 + enumFrom = iterate ((+)1) + enumFromThen n m = iterate ((+)(m-n)) n + toEnum n = fromIntegral n :% 1 + fromEnum = fromInteger . truncate + +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 = 4 :: Int + +-- 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: r*10^e == original r + norm 0 r e = (r, e) + norm ee r e = + let n = 10^ee + tn = 10^n + in if r >= tn then norm ee (r/tn) (e+n) else norm (ee-1) r e + +drop0 "" = "" +drop0 (c:cs) = c : reverse (dropWhile (=='0') (reverse cs)) + +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 = + let s = show ((round (r * 10^n))::Integer) + e = e0+1 + in if e > 0 && e < 8 then + take e s ++ "." ++ drop0 (drop e s) + else if e <= 0 && e > -3 then + "0." ++ take (-e) (repeat '0') ++ drop0 s + else + head s : "."++ drop0 (tail s) ++ "e" ++ show e0 +\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} +{-# SPECIALISE fromRat :: + Rational -> Double, + Rational -> Float #-} +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} +--fromRat :: (RealFloat a) => Rational -> a +fromRat x = + if x == 0 then encodeFloat 0 0 -- Handle exceptional cases + else if x < 0 then - fromRat' (-x) -- first. + else 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 = + if p <= minExp then + (x, p) + else if x >= xMax then + scaleRat b minExp xMin xMax (p+1) (x/b) + else if x < xMin then + scaleRat b minExp xMin xMax (p-1) (x*b) + else + (x, p) + +-- Exponentiation with a cache for the most common numbers. +minExpt = 0::Int +maxExpt = 1100::Int +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 = + if i < b then + 0 + else + -- Try squaring the base first to cut down the number of divisions. + let l = 2 * integerLogBase (b*b) i + doDiv :: Integer -> Int -> Int + doDiv i l = if i < b then l else doDiv (i `div` b) (l+1) + in doDiv (i `div` (b^l)) l +\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 = if x < 0 then showParen (p > 6) + (showChar '-' . showPos (-x)) + else showPos x + +showFloat x = showString (formatRealFloat FFGeneric Nothing x) + +-- These are the format types. This type is not exported. + +data FFFormat = FFExponent | FFFixed | FFGeneric --no need: deriving (Eq, Ord, Show) + +formatRealFloat :: (RealFloat a) => FFFormat -> Maybe Int -> a -> String +formatRealFloat fmt decs x = s + where + base = 10 + s = if isNaN x + then "NaN" + else + if isInfinite x then + if x < 0 then "-Infinity" else "Infinity" + else + if x < 0 || isNegativeZero x then + '-':doFmt fmt (floatToDigits (toInteger base) (-x)) + else + doFmt fmt (floatToDigits (toInteger base) x) + + doFmt fmt (is, e) = + let ds = map intToDigit is in + case fmt of + FFGeneric -> + doFmt (if e <0 || e > 7 then FFExponent else FFFixed) + (is,e) + FFExponent -> + case decs of + Nothing -> + let e' = if e==0 then 0 else e-1 in + (case ds of + [d] -> d : ".0e" + (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 ds = mk0 (reverse s) ++ '.':mk0 ds + f n s "" = f (n-1) ('0':s) "" + f n s (d:ds) = f (n-1) (d:s) ds + in + f e "" ds + Just dec -> + let dec' = max dec 1 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 = + let + v = f d is + in + case v of + (0,is) -> v + (1,is) -> (1, 1:is) + where + b2 = base `div` 2 + + f n [] = (0, replicate n 0) + f 0 (i:_) = (if i>=b2 then 1 else 0, []) + f d (i:is) = + let + (c,ds) = f (d-1) is + i' = c + i + in + if i' == base then (1,0:ds) else (0,i':ds) + +-- +-- 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)) / + 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 (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 (F# x) = F# (negateFloat# x) + +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 (F# x) = I# (float2Int# x) +int2Float (I# x) = F# (int2Float# x) + +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 (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 (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 (D# x) = D# (negateDouble# x) + +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 (D# x) = I# (double2Int# x) +int2Double (I# x) = D# (int2Double# x) +double2Float (D# x) = F# (double2Float# x) +float2Double (F# x) = D# (float2Double# x) + +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 (D# x) (D# y) = D# (x **## y) +\end{code}