2 {-# OPTIONS_GHC -XNoImplicitPrelude #-}
3 {-# OPTIONS_GHC -fno-warn-orphans #-}
4 {-# OPTIONS_HADDOCK hide #-}
5 -----------------------------------------------------------------------------
8 -- Copyright : (c) The University of Glasgow 1994-2002
9 -- License : see libraries/base/LICENSE
11 -- Maintainer : cvs-ghc@haskell.org
12 -- Stability : internal
13 -- Portability : non-portable (GHC Extensions)
15 -- The types 'Float' and 'Double', and the classes 'Floating' and 'RealFloat'.
17 -----------------------------------------------------------------------------
19 #include "ieee-flpt.h"
22 module GHC.Float( module GHC.Float, Float(..), Double(..), Float#, Double# )
39 %*********************************************************
41 \subsection{Standard numeric classes}
43 %*********************************************************
46 -- | Trigonometric and hyperbolic functions and related functions.
48 -- Minimal complete definition:
49 -- 'pi', 'exp', 'log', 'sin', 'cos', 'sinh', 'cosh',
50 -- 'asin', 'acos', 'atan', 'asinh', 'acosh' and 'atanh'
51 class (Fractional a) => Floating a where
53 exp, log, sqrt :: a -> a
54 (**), logBase :: a -> a -> a
55 sin, cos, tan :: a -> a
56 asin, acos, atan :: a -> a
57 sinh, cosh, tanh :: a -> a
58 asinh, acosh, atanh :: a -> a
61 {-# INLINE logBase #-}
65 x ** y = exp (log x * y)
66 logBase x y = log y / log x
69 tanh x = sinh x / cosh x
71 -- | Efficient, machine-independent access to the components of a
72 -- floating-point number.
74 -- Minimal complete definition:
75 -- all except 'exponent', 'significand', 'scaleFloat' and 'atan2'
76 class (RealFrac a, Floating a) => RealFloat a where
77 -- | a constant function, returning the radix of the representation
79 floatRadix :: a -> Integer
80 -- | a constant function, returning the number of digits of
81 -- 'floatRadix' in the significand
82 floatDigits :: a -> Int
83 -- | a constant function, returning the lowest and highest values
84 -- the exponent may assume
85 floatRange :: a -> (Int,Int)
86 -- | The function 'decodeFloat' applied to a real floating-point
87 -- number returns the significand expressed as an 'Integer' and an
88 -- appropriately scaled exponent (an 'Int'). If @'decodeFloat' x@
89 -- yields @(m,n)@, then @x@ is equal in value to @m*b^^n@, where @b@
90 -- is the floating-point radix, and furthermore, either @m@ and @n@
91 -- are both zero or else @b^(d-1) <= m < b^d@, where @d@ is the value
92 -- of @'floatDigits' x@. In particular, @'decodeFloat' 0 = (0,0)@.
93 decodeFloat :: a -> (Integer,Int)
94 -- | 'encodeFloat' performs the inverse of 'decodeFloat'
95 encodeFloat :: Integer -> Int -> a
96 -- | the second component of 'decodeFloat'.
98 -- | the first component of 'decodeFloat', scaled to lie in the open
99 -- interval (@-1@,@1@)
100 significand :: a -> a
101 -- | multiplies a floating-point number by an integer power of the radix
102 scaleFloat :: Int -> a -> a
103 -- | 'True' if the argument is an IEEE \"not-a-number\" (NaN) value
105 -- | 'True' if the argument is an IEEE infinity or negative infinity
106 isInfinite :: a -> Bool
107 -- | 'True' if the argument is too small to be represented in
109 isDenormalized :: a -> Bool
110 -- | 'True' if the argument is an IEEE negative zero
111 isNegativeZero :: a -> Bool
112 -- | 'True' if the argument is an IEEE floating point number
114 -- | a version of arctangent taking two real floating-point arguments.
115 -- For real floating @x@ and @y@, @'atan2' y x@ computes the angle
116 -- (from the positive x-axis) of the vector from the origin to the
117 -- point @(x,y)@. @'atan2' y x@ returns a value in the range [@-pi@,
118 -- @pi@]. It follows the Common Lisp semantics for the origin when
119 -- signed zeroes are supported. @'atan2' y 1@, with @y@ in a type
120 -- that is 'RealFloat', should return the same value as @'atan' y@.
121 -- A default definition of 'atan2' is provided, but implementors
122 -- can provide a more accurate implementation.
126 exponent x = if m == 0 then 0 else n + floatDigits x
127 where (m,n) = decodeFloat x
129 significand x = encodeFloat m (negate (floatDigits x))
130 where (m,_) = decodeFloat x
132 scaleFloat k x = encodeFloat m (n+k)
133 where (m,n) = decodeFloat x
137 | x == 0 && y > 0 = pi/2
138 | x < 0 && y > 0 = pi + atan (y/x)
139 |(x <= 0 && y < 0) ||
140 (x < 0 && isNegativeZero y) ||
141 (isNegativeZero x && isNegativeZero y)
143 | y == 0 && (x < 0 || isNegativeZero x)
144 = pi -- must be after the previous test on zero y
145 | x==0 && y==0 = y -- must be after the other double zero tests
146 | otherwise = x + y -- x or y is a NaN, return a NaN (via +)
150 %*********************************************************
152 \subsection{Type @Float@}
154 %*********************************************************
157 instance Eq Float where
158 (F# x) == (F# y) = x `eqFloat#` y
160 instance Ord Float where
161 (F# x) `compare` (F# y) | x `ltFloat#` y = LT
162 | x `eqFloat#` y = EQ
165 (F# x) < (F# y) = x `ltFloat#` y
166 (F# x) <= (F# y) = x `leFloat#` y
167 (F# x) >= (F# y) = x `geFloat#` y
168 (F# x) > (F# y) = x `gtFloat#` y
170 instance Num Float where
171 (+) x y = plusFloat x y
172 (-) x y = minusFloat x y
173 negate x = negateFloat x
174 (*) x y = timesFloat x y
176 | otherwise = negateFloat x
177 signum x | x == 0.0 = 0
179 | otherwise = negate 1
181 {-# INLINE fromInteger #-}
182 fromInteger i = F# (floatFromInteger i)
184 instance Real Float where
185 toRational x = (m%1)*(b%1)^^n
186 where (m,n) = decodeFloat x
189 instance Fractional Float where
190 (/) x y = divideFloat x y
191 fromRational x = fromRat x
194 {-# RULES "truncate/Float->Int" truncate = float2Int #-}
195 instance RealFrac Float where
197 {-# SPECIALIZE properFraction :: Float -> (Int, Float) #-}
198 {-# SPECIALIZE round :: Float -> Int #-}
200 {-# SPECIALIZE properFraction :: Float -> (Integer, Float) #-}
201 {-# SPECIALIZE round :: Float -> Integer #-}
203 -- ceiling, floor, and truncate are all small
204 {-# INLINE ceiling #-}
206 {-# INLINE truncate #-}
208 -- We assume that FLT_RADIX is 2 so that we can use more efficient code
210 #error FLT_RADIX must be 2
212 properFraction (F# x#)
213 = case decodeFloat_Int# x# of
219 then (fromIntegral m * (2 ^ n), 0.0)
220 else let i = if m >= 0 then m `shiftR` negate n
221 else negate (negate m `shiftR` negate n)
222 f = m - (i `shiftL` negate n)
223 in (fromIntegral i, encodeFloat (fromIntegral f) n)
225 truncate x = case properFraction x of
228 round x = case properFraction x of
230 m = if r < 0.0 then n - 1 else n + 1
231 half_down = abs r - 0.5
233 case (compare half_down 0.0) of
235 EQ -> if even n then n else m
238 ceiling x = case properFraction x of
239 (n,r) -> if r > 0.0 then n + 1 else n
241 floor x = case properFraction x of
242 (n,r) -> if r < 0.0 then n - 1 else n
244 instance Floating Float where
245 pi = 3.141592653589793238
258 (**) x y = powerFloat x y
259 logBase x y = log y / log x
261 asinh x = log (x + sqrt (1.0+x*x))
262 acosh x = log (x + (x+1.0) * sqrt ((x-1.0)/(x+1.0)))
263 atanh x = log ((x+1.0) / sqrt (1.0-x*x))
265 instance RealFloat Float where
266 floatRadix _ = FLT_RADIX -- from float.h
267 floatDigits _ = FLT_MANT_DIG -- ditto
268 floatRange _ = (FLT_MIN_EXP, FLT_MAX_EXP) -- ditto
270 decodeFloat (F# f#) = case decodeFloat_Int# f# of
271 (# i, e #) -> (smallInteger i, I# e)
273 encodeFloat i (I# e) = F# (encodeFloatInteger i e)
275 exponent x = case decodeFloat x of
276 (m,n) -> if m == 0 then 0 else n + floatDigits x
278 significand x = case decodeFloat x of
279 (m,_) -> encodeFloat m (negate (floatDigits x))
281 scaleFloat k x = case decodeFloat x of
282 (m,n) -> encodeFloat m (n+k)
283 isNaN x = 0 /= isFloatNaN x
284 isInfinite x = 0 /= isFloatInfinite x
285 isDenormalized x = 0 /= isFloatDenormalized x
286 isNegativeZero x = 0 /= isFloatNegativeZero x
289 instance Show Float where
290 showsPrec x = showSignedFloat showFloat x
291 showList = showList__ (showsPrec 0)
294 %*********************************************************
296 \subsection{Type @Double@}
298 %*********************************************************
301 instance Eq Double where
302 (D# x) == (D# y) = x ==## y
304 instance Ord Double where
305 (D# x) `compare` (D# y) | x <## y = LT
309 (D# x) < (D# y) = x <## y
310 (D# x) <= (D# y) = x <=## y
311 (D# x) >= (D# y) = x >=## y
312 (D# x) > (D# y) = x >## y
314 instance Num Double where
315 (+) x y = plusDouble x y
316 (-) x y = minusDouble x y
317 negate x = negateDouble x
318 (*) x y = timesDouble x y
320 | otherwise = negateDouble x
321 signum x | x == 0.0 = 0
323 | otherwise = negate 1
325 {-# INLINE fromInteger #-}
326 fromInteger i = D# (doubleFromInteger i)
329 instance Real Double where
330 toRational x = (m%1)*(b%1)^^n
331 where (m,n) = decodeFloat x
334 instance Fractional Double where
335 (/) x y = divideDouble x y
336 fromRational x = fromRat x
339 instance Floating Double where
340 pi = 3.141592653589793238
343 sqrt x = sqrtDouble x
347 asin x = asinDouble x
348 acos x = acosDouble x
349 atan x = atanDouble x
350 sinh x = sinhDouble x
351 cosh x = coshDouble x
352 tanh x = tanhDouble x
353 (**) x y = powerDouble x y
354 logBase x y = log y / log x
356 asinh x = log (x + sqrt (1.0+x*x))
357 acosh x = log (x + (x+1.0) * sqrt ((x-1.0)/(x+1.0)))
358 atanh x = log ((x+1.0) / sqrt (1.0-x*x))
360 {-# RULES "truncate/Double->Int" truncate = double2Int #-}
361 instance RealFrac Double where
363 {-# SPECIALIZE properFraction :: Double -> (Int, Double) #-}
364 {-# SPECIALIZE round :: Double -> Int #-}
366 {-# SPECIALIZE properFraction :: Double -> (Integer, Double) #-}
367 {-# SPECIALIZE round :: Double -> Integer #-}
369 -- ceiling, floor, and truncate are all small
370 {-# INLINE ceiling #-}
372 {-# INLINE truncate #-}
375 = case (decodeFloat x) of { (m,n) ->
376 let b = floatRadix x in
378 (fromInteger m * fromInteger b ^ n, 0.0)
380 case (quotRem m (b^(negate n))) of { (w,r) ->
381 (fromInteger w, encodeFloat r n)
385 truncate x = case properFraction x of
388 round x = case properFraction x of
390 m = if r < 0.0 then n - 1 else n + 1
391 half_down = abs r - 0.5
393 case (compare half_down 0.0) of
395 EQ -> if even n then n else m
398 ceiling x = case properFraction x of
399 (n,r) -> if r > 0.0 then n + 1 else n
401 floor x = case properFraction x of
402 (n,r) -> if r < 0.0 then n - 1 else n
404 instance RealFloat Double where
405 floatRadix _ = FLT_RADIX -- from float.h
406 floatDigits _ = DBL_MANT_DIG -- ditto
407 floatRange _ = (DBL_MIN_EXP, DBL_MAX_EXP) -- ditto
410 = case decodeDoubleInteger x# of
411 (# i, j #) -> (i, I# j)
413 encodeFloat i (I# j) = D# (encodeDoubleInteger i j)
415 exponent x = case decodeFloat x of
416 (m,n) -> if m == 0 then 0 else n + floatDigits x
418 significand x = case decodeFloat x of
419 (m,_) -> encodeFloat m (negate (floatDigits x))
421 scaleFloat k x = case decodeFloat x of
422 (m,n) -> encodeFloat m (n+k)
424 isNaN x = 0 /= isDoubleNaN x
425 isInfinite x = 0 /= isDoubleInfinite x
426 isDenormalized x = 0 /= isDoubleDenormalized x
427 isNegativeZero x = 0 /= isDoubleNegativeZero x
430 instance Show Double where
431 showsPrec x = showSignedFloat showFloat x
432 showList = showList__ (showsPrec 0)
435 %*********************************************************
437 \subsection{@Enum@ instances}
439 %*********************************************************
441 The @Enum@ instances for Floats and Doubles are slightly unusual.
442 The @toEnum@ function truncates numbers to Int. The definitions
443 of @enumFrom@ and @enumFromThen@ allow floats to be used in arithmetic
444 series: [0,0.1 .. 1.0]. However, roundoff errors make these somewhat
445 dubious. This example may have either 10 or 11 elements, depending on
446 how 0.1 is represented.
448 NOTE: The instances for Float and Double do not make use of the default
449 methods for @enumFromTo@ and @enumFromThenTo@, as these rely on there being
450 a `non-lossy' conversion to and from Ints. Instead we make use of the
451 1.2 default methods (back in the days when Enum had Ord as a superclass)
452 for these (@numericEnumFromTo@ and @numericEnumFromThenTo@ below.)
455 instance Enum Float where
459 fromEnum = fromInteger . truncate -- may overflow
460 enumFrom = numericEnumFrom
461 enumFromTo = numericEnumFromTo
462 enumFromThen = numericEnumFromThen
463 enumFromThenTo = numericEnumFromThenTo
465 instance Enum Double where
469 fromEnum = fromInteger . truncate -- may overflow
470 enumFrom = numericEnumFrom
471 enumFromTo = numericEnumFromTo
472 enumFromThen = numericEnumFromThen
473 enumFromThenTo = numericEnumFromThenTo
477 %*********************************************************
479 \subsection{Printing floating point}
481 %*********************************************************
485 -- | Show a signed 'RealFloat' value to full precision
486 -- using standard decimal notation for arguments whose absolute value lies
487 -- between @0.1@ and @9,999,999@, and scientific notation otherwise.
488 showFloat :: (RealFloat a) => a -> ShowS
489 showFloat x = showString (formatRealFloat FFGeneric Nothing x)
491 -- These are the format types. This type is not exported.
493 data FFFormat = FFExponent | FFFixed | FFGeneric
495 formatRealFloat :: (RealFloat a) => FFFormat -> Maybe Int -> a -> String
496 formatRealFloat fmt decs x
498 | isInfinite x = if x < 0 then "-Infinity" else "Infinity"
499 | x < 0 || isNegativeZero x = '-':doFmt fmt (floatToDigits (toInteger base) (-x))
500 | otherwise = doFmt fmt (floatToDigits (toInteger base) x)
504 doFmt format (is, e) =
505 let ds = map intToDigit is in
508 doFmt (if e < 0 || e > 7 then FFExponent else FFFixed)
513 let show_e' = show (e-1) in
516 [d] -> d : ".0e" ++ show_e'
517 (d:ds') -> d : '.' : ds' ++ "e" ++ show_e'
518 [] -> error "formatRealFloat/doFmt/FFExponent: []"
520 let dec' = max dec 1 in
522 [0] -> '0' :'.' : take dec' (repeat '0') ++ "e0"
525 (ei,is') = roundTo base (dec'+1) is
526 (d:ds') = map intToDigit (if ei > 0 then init is' else is')
528 d:'.':ds' ++ 'e':show (e-1+ei)
531 mk0 ls = case ls of { "" -> "0" ; _ -> ls}
535 | e <= 0 -> "0." ++ replicate (-e) '0' ++ ds
538 f 0 s rs = mk0 (reverse s) ++ '.':mk0 rs
539 f n s "" = f (n-1) ('0':s) ""
540 f n s (r:rs) = f (n-1) (r:s) rs
544 let dec' = max dec 0 in
547 (ei,is') = roundTo base (dec' + e) is
548 (ls,rs) = splitAt (e+ei) (map intToDigit is')
550 mk0 ls ++ (if null rs then "" else '.':rs)
553 (ei,is') = roundTo base dec' (replicate (-e) 0 ++ is)
554 d:ds' = map intToDigit (if ei > 0 then is' else 0:is')
556 d : (if null ds' then "" else '.':ds')
559 roundTo :: Int -> Int -> [Int] -> (Int,[Int])
564 _ -> error "roundTo: bad Value"
568 f n [] = (0, replicate n 0)
569 f 0 (x:_) = (if x >= b2 then 1 else 0, [])
571 | i' == base = (1,0:ds)
572 | otherwise = (0,i':ds)
577 -- Based on "Printing Floating-Point Numbers Quickly and Accurately"
578 -- by R.G. Burger and R.K. Dybvig in PLDI 96.
579 -- This version uses a much slower logarithm estimator. It should be improved.
581 -- | 'floatToDigits' takes a base and a non-negative 'RealFloat' number,
582 -- and returns a list of digits and an exponent.
583 -- In particular, if @x>=0@, and
585 -- > floatToDigits base x = ([d1,d2,...,dn], e)
591 -- (2) @x = 0.d1d2...dn * (base**e)@
593 -- (3) @0 <= di <= base-1@
595 floatToDigits :: (RealFloat a) => Integer -> a -> ([Int], Int)
596 floatToDigits _ 0 = ([0], 0)
597 floatToDigits base x =
599 (f0, e0) = decodeFloat x
600 (minExp0, _) = floatRange x
603 minExp = minExp0 - p -- the real minimum exponent
604 -- Haskell requires that f be adjusted so denormalized numbers
605 -- will have an impossibly low exponent. Adjust for this.
607 let n = minExp - e0 in
608 if n > 0 then (f0 `div` (b^n), e0+n) else (f0, e0)
613 (f*be*b*2, 2*b, be*b, b)
617 if e > minExp && f == b^(p-1) then
618 (f*b*2, b^(-e+1)*2, b, 1)
620 (f*2, b^(-e)*2, 1, 1)
626 if b == 2 && base == 10 then
627 -- logBase 10 2 is slightly bigger than 3/10 so
628 -- the following will err on the low side. Ignoring
629 -- the fraction will make it err even more.
630 -- Haskell promises that p-1 <= logBase b f < p.
631 (p - 1 + e0) * 3 `div` 10
633 -- f :: Integer, log :: Float -> Float,
634 -- ceiling :: Float -> Int
635 ceiling ((log (fromInteger (f+1) :: Float) +
636 fromIntegral e * log (fromInteger b)) /
637 log (fromInteger base))
638 --WAS: fromInt e * log (fromInteger b))
642 if r + mUp <= expt base n * s then n else fixup (n+1)
644 if expt base (-n) * (r + mUp) <= s then n else fixup (n+1)
648 gen ds rn sN mUpN mDnN =
650 (dn, rn') = (rn * base) `divMod` sN
654 case (rn' < mDnN', rn' + mUpN' > sN) of
655 (True, False) -> dn : ds
656 (False, True) -> dn+1 : ds
657 (True, True) -> if rn' * 2 < sN then dn : ds else dn+1 : ds
658 (False, False) -> gen (dn:ds) rn' sN mUpN' mDnN'
662 gen [] r (s * expt base k) mUp mDn
664 let bk = expt base (-k) in
665 gen [] (r * bk) s (mUp * bk) (mDn * bk)
667 (map fromIntegral (reverse rds), k)
672 %*********************************************************
674 \subsection{Converting from a Rational to a RealFloat
676 %*********************************************************
678 [In response to a request for documentation of how fromRational works,
679 Joe Fasel writes:] A quite reasonable request! This code was added to
680 the Prelude just before the 1.2 release, when Lennart, working with an
681 early version of hbi, noticed that (read . show) was not the identity
682 for floating-point numbers. (There was a one-bit error about half the
683 time.) The original version of the conversion function was in fact
684 simply a floating-point divide, as you suggest above. The new version
685 is, I grant you, somewhat denser.
687 Unfortunately, Joe's code doesn't work! Here's an example:
689 main = putStr (shows (1.82173691287639817263897126389712638972163e-300::Double) "\n")
694 1.8217369128763981e-300
699 fromRat :: (RealFloat a) => Rational -> a
703 -- If the exponent of the nearest floating-point number to x
704 -- is e, then the significand is the integer nearest xb^(-e),
705 -- where b is the floating-point radix. We start with a good
706 -- guess for e, and if it is correct, the exponent of the
707 -- floating-point number we construct will again be e. If
708 -- not, one more iteration is needed.
710 f e = if e' == e then y else f e'
711 where y = encodeFloat (round (x * (1 % b)^^e)) e
712 (_,e') = decodeFloat y
715 -- We obtain a trial exponent by doing a floating-point
716 -- division of x's numerator by its denominator. The
717 -- result of this division may not itself be the ultimate
718 -- result, because of an accumulation of three rounding
721 (s,e) = decodeFloat (fromInteger (numerator x) `asTypeOf` x'
722 / fromInteger (denominator x))
725 Now, here's Lennart's code (which works)
728 -- | Converts a 'Rational' value into any type in class 'RealFloat'.
729 {-# SPECIALISE fromRat :: Rational -> Double,
730 Rational -> Float #-}
731 fromRat :: (RealFloat a) => Rational -> a
733 -- Deal with special cases first, delegating the real work to fromRat'
734 fromRat (n :% 0) | n > 0 = 1/0 -- +Infinity
735 | n < 0 = -1/0 -- -Infinity
736 | otherwise = 0/0 -- NaN
738 fromRat (n :% d) | n > 0 = fromRat' (n :% d)
739 | n < 0 = - fromRat' ((-n) :% d)
740 | otherwise = encodeFloat 0 0 -- Zero
742 -- Conversion process:
743 -- Scale the rational number by the RealFloat base until
744 -- it lies in the range of the mantissa (as used by decodeFloat/encodeFloat).
745 -- Then round the rational to an Integer and encode it with the exponent
746 -- that we got from the scaling.
747 -- To speed up the scaling process we compute the log2 of the number to get
748 -- a first guess of the exponent.
750 fromRat' :: (RealFloat a) => Rational -> a
751 -- Invariant: argument is strictly positive
753 where b = floatRadix r
755 (minExp0, _) = floatRange r
756 minExp = minExp0 - p -- the real minimum exponent
757 xMin = toRational (expt b (p-1))
758 xMax = toRational (expt b p)
759 p0 = (integerLogBase b (numerator x) - integerLogBase b (denominator x) - p) `max` minExp
760 f = if p0 < 0 then 1 % expt b (-p0) else expt b p0 % 1
761 (x', p') = scaleRat (toRational b) minExp xMin xMax p0 (x / f)
762 r = encodeFloat (round x') p'
764 -- Scale x until xMin <= x < xMax, or p (the exponent) <= minExp.
765 scaleRat :: Rational -> Int -> Rational -> Rational -> Int -> Rational -> (Rational, Int)
766 scaleRat b minExp xMin xMax p x
767 | p <= minExp = (x, p)
768 | x >= xMax = scaleRat b minExp xMin xMax (p+1) (x/b)
769 | x < xMin = scaleRat b minExp xMin xMax (p-1) (x*b)
772 -- Exponentiation with a cache for the most common numbers.
773 minExpt, maxExpt :: Int
777 expt :: Integer -> Int -> Integer
779 if base == 2 && n >= minExpt && n <= maxExpt then
784 expts :: Array Int Integer
785 expts = array (minExpt,maxExpt) [(n,2^n) | n <- [minExpt .. maxExpt]]
787 -- Compute the (floor of the) log of i in base b.
788 -- Simplest way would be just divide i by b until it's smaller then b, but that would
789 -- be very slow! We are just slightly more clever.
790 integerLogBase :: Integer -> Integer -> Int
793 | otherwise = doDiv (i `div` (b^l)) l
795 -- Try squaring the base first to cut down the number of divisions.
796 l = 2 * integerLogBase (b*b) i
798 doDiv :: Integer -> Int -> Int
801 | otherwise = doDiv (x `div` b) (y+1)
806 %*********************************************************
808 \subsection{Floating point numeric primops}
810 %*********************************************************
812 Definitions of the boxed PrimOps; these will be
813 used in the case of partial applications, etc.
816 plusFloat, minusFloat, timesFloat, divideFloat :: Float -> Float -> Float
817 plusFloat (F# x) (F# y) = F# (plusFloat# x y)
818 minusFloat (F# x) (F# y) = F# (minusFloat# x y)
819 timesFloat (F# x) (F# y) = F# (timesFloat# x y)
820 divideFloat (F# x) (F# y) = F# (divideFloat# x y)
822 negateFloat :: Float -> Float
823 negateFloat (F# x) = F# (negateFloat# x)
825 gtFloat, geFloat, eqFloat, neFloat, ltFloat, leFloat :: Float -> Float -> Bool
826 gtFloat (F# x) (F# y) = gtFloat# x y
827 geFloat (F# x) (F# y) = geFloat# x y
828 eqFloat (F# x) (F# y) = eqFloat# x y
829 neFloat (F# x) (F# y) = neFloat# x y
830 ltFloat (F# x) (F# y) = ltFloat# x y
831 leFloat (F# x) (F# y) = leFloat# x y
833 float2Int :: Float -> Int
834 float2Int (F# x) = I# (float2Int# x)
836 int2Float :: Int -> Float
837 int2Float (I# x) = F# (int2Float# x)
839 expFloat, logFloat, sqrtFloat :: Float -> Float
840 sinFloat, cosFloat, tanFloat :: Float -> Float
841 asinFloat, acosFloat, atanFloat :: Float -> Float
842 sinhFloat, coshFloat, tanhFloat :: Float -> Float
843 expFloat (F# x) = F# (expFloat# x)
844 logFloat (F# x) = F# (logFloat# x)
845 sqrtFloat (F# x) = F# (sqrtFloat# x)
846 sinFloat (F# x) = F# (sinFloat# x)
847 cosFloat (F# x) = F# (cosFloat# x)
848 tanFloat (F# x) = F# (tanFloat# x)
849 asinFloat (F# x) = F# (asinFloat# x)
850 acosFloat (F# x) = F# (acosFloat# x)
851 atanFloat (F# x) = F# (atanFloat# x)
852 sinhFloat (F# x) = F# (sinhFloat# x)
853 coshFloat (F# x) = F# (coshFloat# x)
854 tanhFloat (F# x) = F# (tanhFloat# x)
856 powerFloat :: Float -> Float -> Float
857 powerFloat (F# x) (F# y) = F# (powerFloat# x y)
859 -- definitions of the boxed PrimOps; these will be
860 -- used in the case of partial applications, etc.
862 plusDouble, minusDouble, timesDouble, divideDouble :: Double -> Double -> Double
863 plusDouble (D# x) (D# y) = D# (x +## y)
864 minusDouble (D# x) (D# y) = D# (x -## y)
865 timesDouble (D# x) (D# y) = D# (x *## y)
866 divideDouble (D# x) (D# y) = D# (x /## y)
868 negateDouble :: Double -> Double
869 negateDouble (D# x) = D# (negateDouble# x)
871 gtDouble, geDouble, eqDouble, neDouble, leDouble, ltDouble :: Double -> Double -> Bool
872 gtDouble (D# x) (D# y) = x >## y
873 geDouble (D# x) (D# y) = x >=## y
874 eqDouble (D# x) (D# y) = x ==## y
875 neDouble (D# x) (D# y) = x /=## y
876 ltDouble (D# x) (D# y) = x <## y
877 leDouble (D# x) (D# y) = x <=## y
879 double2Int :: Double -> Int
880 double2Int (D# x) = I# (double2Int# x)
882 int2Double :: Int -> Double
883 int2Double (I# x) = D# (int2Double# x)
885 double2Float :: Double -> Float
886 double2Float (D# x) = F# (double2Float# x)
888 float2Double :: Float -> Double
889 float2Double (F# x) = D# (float2Double# x)
891 expDouble, logDouble, sqrtDouble :: Double -> Double
892 sinDouble, cosDouble, tanDouble :: Double -> Double
893 asinDouble, acosDouble, atanDouble :: Double -> Double
894 sinhDouble, coshDouble, tanhDouble :: Double -> Double
895 expDouble (D# x) = D# (expDouble# x)
896 logDouble (D# x) = D# (logDouble# x)
897 sqrtDouble (D# x) = D# (sqrtDouble# x)
898 sinDouble (D# x) = D# (sinDouble# x)
899 cosDouble (D# x) = D# (cosDouble# x)
900 tanDouble (D# x) = D# (tanDouble# x)
901 asinDouble (D# x) = D# (asinDouble# x)
902 acosDouble (D# x) = D# (acosDouble# x)
903 atanDouble (D# x) = D# (atanDouble# x)
904 sinhDouble (D# x) = D# (sinhDouble# x)
905 coshDouble (D# x) = D# (coshDouble# x)
906 tanhDouble (D# x) = D# (tanhDouble# x)
908 powerDouble :: Double -> Double -> Double
909 powerDouble (D# x) (D# y) = D# (x **## y)
913 foreign import ccall unsafe "isFloatNaN" isFloatNaN :: Float -> Int
914 foreign import ccall unsafe "isFloatInfinite" isFloatInfinite :: Float -> Int
915 foreign import ccall unsafe "isFloatDenormalized" isFloatDenormalized :: Float -> Int
916 foreign import ccall unsafe "isFloatNegativeZero" isFloatNegativeZero :: Float -> Int
919 foreign import ccall unsafe "isDoubleNaN" isDoubleNaN :: Double -> Int
920 foreign import ccall unsafe "isDoubleInfinite" isDoubleInfinite :: Double -> Int
921 foreign import ccall unsafe "isDoubleDenormalized" isDoubleDenormalized :: Double -> Int
922 foreign import ccall unsafe "isDoubleNegativeZero" isDoubleNegativeZero :: Double -> Int
925 %*********************************************************
927 \subsection{Coercion rules}
929 %*********************************************************
933 "fromIntegral/Int->Float" fromIntegral = int2Float
934 "fromIntegral/Int->Double" fromIntegral = int2Double
935 "realToFrac/Float->Float" realToFrac = id :: Float -> Float
936 "realToFrac/Float->Double" realToFrac = float2Double
937 "realToFrac/Double->Float" realToFrac = double2Float
938 "realToFrac/Double->Double" realToFrac = id :: Double -> Double
939 "realToFrac/Int->Double" realToFrac = int2Double -- See Note [realToFrac int-to-float]
940 "realToFrac/Int->Float" realToFrac = int2Float -- ..ditto
944 Note [realToFrac int-to-float]
945 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
946 Don found that the RULES for realToFrac/Int->Double and simliarly
947 Float made a huge difference to some stream-fusion programs. Here's
950 import Data.Array.Vector
955 let c = replicateU n (2::Double)
956 a = mapU realToFrac (enumFromToU 0 (n-1) ) :: UArr Double
957 print (sumU (zipWithU (*) c a))
959 Without the RULE we get this loop body:
961 case $wtoRational sc_sY4 of ww_aM7 { (# ww1_aM9, ww2_aMa #) ->
962 case $wfromRat ww1_aM9 ww2_aMa of tpl_X1P { D# ipv_sW3 ->
966 (+## sc2_sY6 (*## 2.0 ipv_sW3))
973 (+## sc2_sXV (*## 2.0 (int2Double# sc_sXT)))
975 The running time of the program goes from 120 seconds to 0.198 seconds
976 with the native backend, and 0.143 seconds with the C backend.
978 A few more details in Trac #2251, and the patch message
979 "Add RULES for realToFrac from Int".
981 %*********************************************************
985 %*********************************************************
988 showSignedFloat :: (RealFloat a)
989 => (a -> ShowS) -- ^ a function that can show unsigned values
990 -> Int -- ^ the precedence of the enclosing context
991 -> a -- ^ the value to show
993 showSignedFloat showPos p x
994 | x < 0 || isNegativeZero x
995 = showParen (p > 6) (showChar '-' . showPos (-x))
996 | otherwise = showPos x