\begin{code}
#include "ieee-flpt.h"
module GHC.Float( module GHC.Float, Float(..), Double(..), Float#, Double# )
where
import Data.Maybe
import Data.Bits
import GHC.Base
import GHC.List
import GHC.Enum
import GHC.Show
import GHC.Num
import GHC.Real
import GHC.Arr
infixr 8 **
\end{code}
%*********************************************************
%* *
\subsection{Standard numeric classes}
%* *
%*********************************************************
\begin{code}
class (Fractional a) => Floating a where
pi :: a
exp, log, sqrt :: a -> a
(**), logBase :: a -> a -> a
sin, cos, tan :: a -> a
asin, acos, atan :: a -> a
sinh, cosh, tanh :: a -> a
asinh, acosh, atanh :: a -> a
x ** y = exp (log x * y)
logBase x y = log y / log x
sqrt x = x ** 0.5
tan x = sin x / cos x
tanh x = sinh x / cosh x
class (RealFrac a, Floating a) => RealFloat a where
floatRadix :: a -> Integer
floatDigits :: a -> Int
floatRange :: a -> (Int,Int)
decodeFloat :: a -> (Integer,Int)
encodeFloat :: Integer -> Int -> a
exponent :: a -> Int
significand :: a -> a
scaleFloat :: Int -> a -> a
isNaN :: a -> Bool
isInfinite :: a -> Bool
isDenormalized :: a -> Bool
isNegativeZero :: a -> Bool
isIEEE :: a -> Bool
atan2 :: a -> a -> a
exponent x = if m == 0 then 0 else n + floatDigits x
where (m,n) = decodeFloat x
significand x = encodeFloat m (negate (floatDigits x))
where (m,_) = decodeFloat x
scaleFloat k x = encodeFloat m (n + clamp b k)
where (m,n) = decodeFloat x
(l,h) = floatRange x
d = floatDigits x
b = h l + 4*d
atan2 y x
| x > 0 = atan (y/x)
| x == 0 && y > 0 = pi/2
| x < 0 && y > 0 = pi + atan (y/x)
|(x <= 0 && y < 0) ||
(x < 0 && isNegativeZero y) ||
(isNegativeZero x && isNegativeZero y)
= atan2 (y) x
| y == 0 && (x < 0 || isNegativeZero x)
= pi
| x==0 && y==0 = y
| otherwise = x + y
\end{code}
%*********************************************************
%* *
\subsection{Type @Float@}
%* *
%*********************************************************
\begin{code}
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 i = F# (floatFromInteger 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 RealFrac Float where
#if FLT_RADIX != 2
#error FLT_RADIX must be 2
#endif
properFraction (F# x#)
= case decodeFloat_Int# x# of
(# m#, n# #) ->
let m = I# m#
n = I# n#
in
if n >= 0
then (fromIntegral m * (2 ^ n), 0.0)
else let i = if m >= 0 then m `shiftR` negate n
else negate (negate m `shiftR` negate n)
f = m (i `shiftL` negate n)
in (fromIntegral i, encodeFloat (fromIntegral f) n)
truncate x = case properFraction x of
(n,_) -> n
round x = case properFraction x of
(n,r) -> let
m = if r < 0.0 then n 1 else n + 1
half_down = abs r 0.5
in
case (compare half_down 0.0) of
LT -> n
EQ -> if even n then n else m
GT -> m
ceiling x = case properFraction x of
(n,r) -> if r > 0.0 then n + 1 else n
floor x = case properFraction x of
(n,r) -> if r < 0.0 then n 1 else n
instance Floating Float where
pi = 3.141592653589793238
exp x = expFloat x
log x = logFloat x
sqrt x = sqrtFloat x
sin x = sinFloat x
cos x = cosFloat x
tan x = tanFloat x
asin x = asinFloat x
acos x = acosFloat x
atan x = atanFloat x
sinh x = sinhFloat x
cosh x = coshFloat x
tanh x = tanhFloat x
(**) x y = powerFloat x y
logBase x y = log y / log x
asinh x = log (x + sqrt (1.0+x*x))
acosh x = log (x + (x+1.0) * sqrt ((x1.0)/(x+1.0)))
atanh x = 0.5 * log ((1.0+x) / (1.0x))
instance RealFloat Float where
floatRadix _ = FLT_RADIX
floatDigits _ = FLT_MANT_DIG
floatRange _ = (FLT_MIN_EXP, FLT_MAX_EXP)
decodeFloat (F# f#) = case decodeFloat_Int# f# of
(# i, e #) -> (smallInteger i, I# e)
encodeFloat i (I# e) = F# (encodeFloatInteger i 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 + clamp bf k)
where bf = FLT_MAX_EXP (FLT_MIN_EXP) + 4*FLT_MANT_DIG
isNaN x = 0 /= isFloatNaN x
isInfinite x = 0 /= isFloatInfinite x
isDenormalized x = 0 /= isFloatDenormalized x
isNegativeZero x = 0 /= isFloatNegativeZero x
isIEEE _ = True
instance Show Float where
showsPrec x = showSignedFloat showFloat x
showList = showList__ (showsPrec 0)
\end{code}
%*********************************************************
%* *
\subsection{Type @Double@}
%* *
%*********************************************************
\begin{code}
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 i = D# (doubleFromInteger i)
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 ((x1.0)/(x+1.0)))
atanh x = 0.5 * log ((1.0+x) / (1.0x))
instance RealFrac Double where
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
floatDigits _ = DBL_MANT_DIG
floatRange _ = (DBL_MIN_EXP, DBL_MAX_EXP)
decodeFloat (D# x#)
= case decodeDoubleInteger x# of
(# i, j #) -> (i, I# j)
encodeFloat i (I# j) = D# (encodeDoubleInteger i j)
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 + clamp bd k)
where bd = DBL_MAX_EXP (DBL_MIN_EXP) + 4*DBL_MANT_DIG
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 = showSignedFloat showFloat x
showList = showList__ (showsPrec 0)
\end{code}
%*********************************************************
%* *
\subsection{@Enum@ instances}
%* *
%*********************************************************
The @Enum@ instances for Floats and Doubles are slightly unusual.
The @toEnum@ function truncates numbers to Int. The definitions
of @enumFrom@ and @enumFromThen@ allow floats to be used in arithmetic
series: [0,0.1 .. 1.0]. However, roundoff errors make these somewhat
dubious. This example may have either 10 or 11 elements, depending on
how 0.1 is represented.
NOTE: The instances for Float and Double do not make use of the default
methods for @enumFromTo@ and @enumFromThenTo@, as these rely on there being
a `non-lossy' conversion to and from Ints. Instead we make use of the
1.2 default methods (back in the days when Enum had Ord as a superclass)
for these (@numericEnumFromTo@ and @numericEnumFromThenTo@ below.)
\begin{code}
instance Enum Float where
succ x = x + 1
pred x = x 1
toEnum = int2Float
fromEnum = fromInteger . truncate
enumFrom = numericEnumFrom
enumFromTo = numericEnumFromTo
enumFromThen = numericEnumFromThen
enumFromThenTo = numericEnumFromThenTo
instance Enum Double where
succ x = x + 1
pred x = x 1
toEnum = int2Double
fromEnum = fromInteger . truncate
enumFrom = numericEnumFrom
enumFromTo = numericEnumFromTo
enumFromThen = numericEnumFromThen
enumFromThenTo = numericEnumFromThenTo
\end{code}
%*********************************************************
%* *
\subsection{Printing floating point}
%* *
%*********************************************************
\begin{code}
showFloat :: (RealFloat a) => a -> ShowS
showFloat x = showString (formatRealFloat FFGeneric Nothing x)
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 (e1) in
case ds of
"0" -> "0.0e0"
[d] -> d : ".0e" ++ show_e'
(d:ds') -> d : '.' : ds' ++ "e" ++ show_e'
[] -> error "formatRealFloat/doFmt/FFExponent: []"
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 (e1+ei)
FFFixed ->
let
mk0 ls = case ls of { "" -> "0" ; _ -> ls}
in
case decs of
Nothing
| e <= 0 -> "0." ++ replicate (e) '0' ++ ds
| otherwise ->
let
f 0 s rs = mk0 (reverse s) ++ '.':mk0 rs
f n s "" = f (n1) ('0':s) ""
f n s (r:rs) = f (n1) (r:s) rs
in
f e "" ds
Just dec ->
let dec' = max dec 0 in
if e >= 0 then
let
(ei,is') = roundTo base (dec' + e) is
(ls,rs) = splitAt (e+ei) (map intToDigit is')
in
mk0 ls ++ (if null rs then "" else '.':rs)
else
let
(ei,is') = roundTo base dec' (replicate (e) 0 ++ is)
d:ds' = map intToDigit (if ei > 0 then is' else 0:is')
in
d : (if null ds' then "" else '.':ds')
roundTo :: Int -> Int -> [Int] -> (Int,[Int])
roundTo base d is =
case f d is of
x@(0,_) -> x
(1,xs) -> (1, 1:xs)
_ -> error "roundTo: bad Value"
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 (n1) xs
i' = c + i
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
(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^(p1) then
(f*be*b*2, 2*b, be*b, be)
else
(f*be*2, 2, be, be)
else
if e > minExp && f == b^(p1) then
(f*b*2, b^(e+1)*2, b, 1)
else
(f*2, b^(e)*2, 1, 1)
k :: Int
k =
let
k0 :: Int
k0 =
if b == 2 && base == 10 then
let lx = p 1 + e0
k1 = (lx * 8651) `quot` 28738
in if lx >= 0 then k1 + 1 else k1
else
ceiling ((log (fromInteger (f+1) :: Float) +
fromIntegral e * log (fromInteger b)) /
log (fromInteger base))
fixup n =
if n >= 0 then
if r + mUp <= expt base n * s then n else fixup (n+1)
else
if expt base (n) * (r + mUp) <= s then n else fixup (n+1)
in
fixup k0
gen ds rn sN mUpN mDnN =
let
(dn, rn') = (rn * base) `divMod` sN
mUpN' = mUpN * base
mDnN' = mDnN * base
in
case (rn' < mDnN', rn' + mUpN' > sN) of
(True, False) -> dn : ds
(False, True) -> dn+1 : ds
(True, True) -> if rn' * 2 < sN then dn : ds else dn+1 : ds
(False, False) -> gen (dn:ds) rn' sN mUpN' mDnN'
rds =
if k >= 0 then
gen [] r (s * expt base k) mUp mDn
else
let bk = expt base (k) in
gen [] (r * bk) s (mUp * bk) (mDn * bk)
in
(map fromIntegral (reverse rds), k)
\end{code}
%*********************************************************
%* *
\subsection{Converting from a Rational to a RealFloat
%* *
%*********************************************************
[In response to a request for documentation of how fromRational works,
Joe Fasel writes:] A quite reasonable request! This code was added to
the Prelude just before the 1.2 release, when Lennart, working with an
early version of hbi, noticed that (read . show) was not the identity
for floating-point numbers. (There was a one-bit error about half the
time.) The original version of the conversion function was in fact
simply a floating-point divide, as you suggest above. The new version
is, I grant you, somewhat denser.
Unfortunately, Joe's code doesn't work! Here's an example:
main = putStr (shows (1.82173691287639817263897126389712638972163e-300::Double) "\n")
This program prints
0.0000000000000000
instead of
1.8217369128763981e-300
Here's Joe's code:
\begin{pseudocode}
fromRat :: (RealFloat a) => Rational -> a
fromRat x = x'
where x' = f e
-- If the exponent of the nearest floating-point number to x
-- is e, then the significand is the integer nearest xb^(-e),
-- where b is the floating-point radix. We start with a good
-- guess for e, and if it is correct, the exponent of the
-- floating-point number we construct will again be e. If
-- not, one more iteration is needed.
f e = if e' == e then y else f e'
where y = encodeFloat (round (x * (1 % b)^^e)) e
(_,e') = decodeFloat y
b = floatRadix x'
-- We obtain a trial exponent by doing a floating-point
-- division of x's numerator by its denominator. The
-- result of this division may not itself be the ultimate
-- result, because of an accumulation of three rounding
-- errors.
(s,e) = decodeFloat (fromInteger (numerator x) `asTypeOf` x'
/ fromInteger (denominator x))
\end{pseudocode}
Now, here's Lennart's code (which works)
\begin{code}
fromRat :: (RealFloat a) => Rational -> a
fromRat (n :% 0) | n > 0 = 1/0
| n < 0 = 1/0
| otherwise = 0/0
fromRat (n :% d) | n > 0 = fromRat' (n :% d)
| n < 0 = fromRat' ((n) :% d)
| otherwise = encodeFloat 0 0
fromRat' :: (RealFloat a) => Rational -> a
fromRat' x = r
where b = floatRadix r
p = floatDigits r
(minExp0, _) = floatRange r
minExp = minExp0 p
xMin = toRational (expt b (p1))
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'
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 (p1) (x*b)
| otherwise = (x, p)
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]]
integerLogBase :: Integer -> Integer -> Int
integerLogBase b i
| i < b = 0
| otherwise = doDiv (i `div` (b^l)) l
where
l = 2 * integerLogBase (b*b) i
doDiv :: Integer -> Int -> Int
doDiv x y
| x < b = y
| otherwise = doDiv (x `div` b) (y+1)
\end{code}
%*********************************************************
%* *
\subsection{Floating point numeric primops}
%* *
%*********************************************************
Definitions of the boxed PrimOps; these will be
used in the case of partial applications, etc.
\begin{code}
plusFloat, minusFloat, timesFloat, divideFloat :: Float -> Float -> Float
plusFloat (F# x) (F# y) = F# (plusFloat# x y)
minusFloat (F# x) (F# y) = F# (minusFloat# x y)
timesFloat (F# x) (F# y) = F# (timesFloat# x y)
divideFloat (F# x) (F# y) = F# (divideFloat# x y)
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)
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}
\begin{code}
foreign import ccall unsafe "isFloatNaN" isFloatNaN :: Float -> Int
foreign import ccall unsafe "isFloatInfinite" isFloatInfinite :: Float -> Int
foreign import ccall unsafe "isFloatDenormalized" isFloatDenormalized :: Float -> Int
foreign import ccall unsafe "isFloatNegativeZero" isFloatNegativeZero :: Float -> Int
foreign import ccall unsafe "isDoubleNaN" isDoubleNaN :: Double -> Int
foreign import ccall unsafe "isDoubleInfinite" isDoubleInfinite :: Double -> Int
foreign import ccall unsafe "isDoubleDenormalized" isDoubleDenormalized :: Double -> Int
foreign import ccall unsafe "isDoubleNegativeZero" isDoubleNegativeZero :: Double -> Int
\end{code}
%*********************************************************
%* *
\subsection{Coercion rules}
%* *
%*********************************************************
\begin{code}
\end{code}
Note [realToFrac int-to-float]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Don found that the RULES for realToFrac/Int->Double and simliarly
Float made a huge difference to some stream-fusion programs. Here's
an example
import Data.Array.Vector
n = 40000000
main = do
let c = replicateU n (2::Double)
a = mapU realToFrac (enumFromToU 0 (n-1) ) :: UArr Double
print (sumU (zipWithU (*) c a))
Without the RULE we get this loop body:
case $wtoRational sc_sY4 of ww_aM7 { (# ww1_aM9, ww2_aMa #) ->
case $wfromRat ww1_aM9 ww2_aMa of tpl_X1P { D# ipv_sW3 ->
Main.$s$wfold
(+# sc_sY4 1)
(+# wild_X1i 1)
(+## sc2_sY6 (*## 2.0 ipv_sW3))
And with the rule:
Main.$s$wfold
(+# sc_sXT 1)
(+# wild_X1h 1)
(+## sc2_sXV (*## 2.0 (int2Double# sc_sXT)))
The running time of the program goes from 120 seconds to 0.198 seconds
with the native backend, and 0.143 seconds with the C backend.
A few more details in Trac #2251, and the patch message
"Add RULES for realToFrac from Int".
%*********************************************************
%* *
\subsection{Utils}
%* *
%*********************************************************
\begin{code}
showSignedFloat :: (RealFloat a)
=> (a -> ShowS)
-> Int
-> a
-> ShowS
showSignedFloat showPos p x
| x < 0 || isNegativeZero x
= showParen (p > 6) (showChar '-' . showPos (x))
| otherwise = showPos x
\end{code}
We need to prevent over/underflow of the exponent in encodeFloat when
called from scaleFloat, hence we clamp the scaling parameter.
We must have a large enough range to cover the maximum difference of
exponents returned by decodeFloat.
\begin{code}
clamp :: Int -> Int -> Int
clamp bd k = max (bd) (min bd k)
\end{code}