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