+++ /dev/null
-%
-% (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}
projects / ghc-hetmet.git / commitdiff