2 % (c) The AQUA Project, Glasgow University, 1994-1996
5 \section[PrelNum]{Module @PrelNum@}
19 {-# OPTIONS -fno-implicit-prelude #-}
21 #include "../../includes/ieee-flpt.h"
23 module PrelFloat( module PrelFloat, Float#, Double# ) where
25 import {-# SOURCE #-} PrelErr
38 %*********************************************************
40 \subsection{Standard numeric classes}
42 %*********************************************************
45 class (Fractional a) => Floating a where
47 exp, log, sqrt :: a -> a
48 (**), logBase :: a -> a -> a
49 sin, cos, tan :: a -> a
50 asin, acos, atan :: a -> a
51 sinh, cosh, tanh :: a -> a
52 asinh, acosh, atanh :: a -> a
54 x ** y = exp (log x * y)
55 logBase x y = log y / log x
58 tanh x = sinh x / cosh x
60 class (RealFrac a, Floating a) => RealFloat a where
61 floatRadix :: a -> Integer
62 floatDigits :: a -> Int
63 floatRange :: a -> (Int,Int)
64 decodeFloat :: a -> (Integer,Int)
65 encodeFloat :: Integer -> Int -> a
68 scaleFloat :: Int -> a -> a
69 isNaN, isInfinite, isDenormalized, isNegativeZero, isIEEE
74 exponent x = if m == 0 then 0 else n + floatDigits x
75 where (m,n) = decodeFloat x
77 significand x = encodeFloat m (negate (floatDigits x))
78 where (m,_) = decodeFloat x
80 scaleFloat k x = encodeFloat m (n+k)
81 where (m,n) = decodeFloat x
85 | x == 0 && y > 0 = pi/2
86 | x < 0 && y > 0 = pi + atan (y/x)
88 (x < 0 && isNegativeZero y) ||
89 (isNegativeZero x && isNegativeZero y)
91 | y == 0 && (x < 0 || isNegativeZero x)
92 = pi -- must be after the previous test on zero y
93 | x==0 && y==0 = y -- must be after the other double zero tests
94 | otherwise = x + y -- x or y is a NaN, return a NaN (via +)
98 %*********************************************************
100 \subsection{Type @Integer@, @Float@, @Double@}
102 %*********************************************************
105 data Float = F# Float#
106 data Double = D# Double#
108 instance CCallable Float
109 instance CReturnable Float
111 instance CCallable Double
112 instance CReturnable Double
116 %*********************************************************
118 \subsection{Type @Float@}
120 %*********************************************************
123 instance Eq Float where
124 (F# x) == (F# y) = x `eqFloat#` y
126 instance Ord Float where
127 (F# x) `compare` (F# y) | x `ltFloat#` y = LT
128 | x `eqFloat#` y = EQ
131 (F# x) < (F# y) = x `ltFloat#` y
132 (F# x) <= (F# y) = x `leFloat#` y
133 (F# x) >= (F# y) = x `geFloat#` y
134 (F# x) > (F# y) = x `gtFloat#` y
136 instance Num Float where
137 (+) x y = plusFloat x y
138 (-) x y = minusFloat x y
139 negate x = negateFloat x
140 (*) x y = timesFloat x y
142 | otherwise = negateFloat x
143 signum x | x == 0.0 = 0
145 | otherwise = negate 1
147 {-# INLINE fromInteger #-}
148 fromInteger n = encodeFloat n 0
149 -- It's important that encodeFloat inlines here, and that
150 -- fromInteger in turn inlines,
151 -- so that if fromInteger is applied to an (S# i) the right thing happens
153 {-# INLINE fromInt #-}
154 fromInt i = int2Float i
156 instance Real Float where
157 toRational x = (m%1)*(b%1)^^n
158 where (m,n) = decodeFloat x
161 instance Fractional Float where
162 (/) x y = divideFloat x y
163 fromRational x = fromRat x
166 {-# RULES "truncate/Float->Int" truncate = float2Int #-}
167 instance RealFrac Float where
169 {-# SPECIALIZE properFraction :: Float -> (Int, Float) #-}
170 {-# SPECIALIZE round :: Float -> Int #-}
171 {-# SPECIALIZE ceiling :: Float -> Int #-}
172 {-# SPECIALIZE floor :: Float -> Int #-}
174 {-# SPECIALIZE properFraction :: Float -> (Integer, Float) #-}
175 {-# SPECIALIZE truncate :: Float -> Integer #-}
176 {-# SPECIALIZE round :: Float -> Integer #-}
177 {-# SPECIALIZE ceiling :: Float -> Integer #-}
178 {-# SPECIALIZE floor :: Float -> Integer #-}
181 = case (decodeFloat x) of { (m,n) ->
182 let b = floatRadix x in
184 (fromInteger m * fromInteger b ^ n, 0.0)
186 case (quotRem m (b^(negate n))) of { (w,r) ->
187 (fromInteger w, encodeFloat r n)
191 truncate x = case properFraction x of
194 round x = case properFraction x of
196 m = if r < 0.0 then n - 1 else n + 1
197 half_down = abs r - 0.5
199 case (compare half_down 0.0) of
201 EQ -> if even n then n else m
204 ceiling x = case properFraction x of
205 (n,r) -> if r > 0.0 then n + 1 else n
207 floor x = case properFraction x of
208 (n,r) -> if r < 0.0 then n - 1 else n
210 instance Floating Float where
211 pi = 3.141592653589793238
224 (**) x y = powerFloat x y
225 logBase x y = log y / log x
227 asinh x = log (x + sqrt (1.0+x*x))
228 acosh x = log (x + (x+1.0) * sqrt ((x-1.0)/(x+1.0)))
229 atanh x = log ((x+1.0) / sqrt (1.0-x*x))
231 instance RealFloat Float where
232 floatRadix _ = FLT_RADIX -- from float.h
233 floatDigits _ = FLT_MANT_DIG -- ditto
234 floatRange _ = (FLT_MIN_EXP, FLT_MAX_EXP) -- ditto
237 = case decodeFloat# f# of
238 (# exp#, s#, d# #) -> (J# s# d#, I# exp#)
240 encodeFloat (S# i) j = int_encodeFloat# i j
241 encodeFloat (J# s# d#) e = encodeFloat# s# d# e
243 exponent x = case decodeFloat x of
244 (m,n) -> if m == 0 then 0 else n + floatDigits x
246 significand x = case decodeFloat x of
247 (m,_) -> encodeFloat m (negate (floatDigits x))
249 scaleFloat k x = case decodeFloat x of
250 (m,n) -> encodeFloat m (n+k)
251 isNaN x = 0 /= isFloatNaN x
252 isInfinite x = 0 /= isFloatInfinite x
253 isDenormalized x = 0 /= isFloatDenormalized x
254 isNegativeZero x = 0 /= isFloatNegativeZero x
257 instance Show Float where
258 showsPrec x = showSigned showFloat x
259 showList = showList__ (showsPrec 0)
262 %*********************************************************
264 \subsection{Type @Double@}
266 %*********************************************************
269 instance Eq Double where
270 (D# x) == (D# y) = x ==## y
272 instance Ord Double where
273 (D# x) `compare` (D# y) | x <## y = LT
277 (D# x) < (D# y) = x <## y
278 (D# x) <= (D# y) = x <=## y
279 (D# x) >= (D# y) = x >=## y
280 (D# x) > (D# y) = x >## y
282 instance Num Double where
283 (+) x y = plusDouble x y
284 (-) x y = minusDouble x y
285 negate x = negateDouble x
286 (*) x y = timesDouble x y
288 | otherwise = negateDouble x
289 signum x | x == 0.0 = 0
291 | otherwise = negate 1
293 {-# INLINE fromInteger #-}
294 -- See comments with Num Float
295 fromInteger n = encodeFloat n 0
296 fromInt (I# n#) = case (int2Double# n#) of { d# -> D# d# }
298 instance Real Double where
299 toRational x = (m%1)*(b%1)^^n
300 where (m,n) = decodeFloat x
303 instance Fractional Double where
304 (/) x y = divideDouble x y
305 fromRational x = fromRat x
308 instance Floating Double where
309 pi = 3.141592653589793238
312 sqrt x = sqrtDouble x
316 asin x = asinDouble x
317 acos x = acosDouble x
318 atan x = atanDouble x
319 sinh x = sinhDouble x
320 cosh x = coshDouble x
321 tanh x = tanhDouble x
322 (**) x y = powerDouble x y
323 logBase x y = log y / log x
325 asinh x = log (x + sqrt (1.0+x*x))
326 acosh x = log (x + (x+1.0) * sqrt ((x-1.0)/(x+1.0)))
327 atanh x = log ((x+1.0) / sqrt (1.0-x*x))
329 {-# RULES "truncate/Double->Int" truncate = double2Int #-}
330 instance RealFrac Double where
332 {-# SPECIALIZE properFraction :: Double -> (Int, Double) #-}
333 {-# SPECIALIZE round :: Double -> Int #-}
334 {-# SPECIALIZE ceiling :: Double -> Int #-}
335 {-# SPECIALIZE floor :: Double -> Int #-}
337 {-# SPECIALIZE properFraction :: Double -> (Integer, Double) #-}
338 {-# SPECIALIZE truncate :: Double -> Integer #-}
339 {-# SPECIALIZE round :: Double -> Integer #-}
340 {-# SPECIALIZE ceiling :: Double -> Integer #-}
341 {-# SPECIALIZE floor :: Double -> Integer #-}
344 = case (decodeFloat x) of { (m,n) ->
345 let b = floatRadix x in
347 (fromInteger m * fromInteger b ^ n, 0.0)
349 case (quotRem m (b^(negate n))) of { (w,r) ->
350 (fromInteger w, encodeFloat r n)
354 truncate x = case properFraction x of
357 round x = case properFraction x of
359 m = if r < 0.0 then n - 1 else n + 1
360 half_down = abs r - 0.5
362 case (compare half_down 0.0) of
364 EQ -> if even n then n else m
367 ceiling x = case properFraction x of
368 (n,r) -> if r > 0.0 then n + 1 else n
370 floor x = case properFraction x of
371 (n,r) -> if r < 0.0 then n - 1 else n
373 instance RealFloat Double where
374 floatRadix _ = FLT_RADIX -- from float.h
375 floatDigits _ = DBL_MANT_DIG -- ditto
376 floatRange _ = (DBL_MIN_EXP, DBL_MAX_EXP) -- ditto
379 = case decodeDouble# x# of
380 (# exp#, s#, d# #) -> (J# s# d#, I# exp#)
382 encodeFloat (S# i) j = int_encodeDouble# i j
383 encodeFloat (J# s# d#) e = encodeDouble# s# d# e
385 exponent x = case decodeFloat x of
386 (m,n) -> if m == 0 then 0 else n + floatDigits x
388 significand x = case decodeFloat x of
389 (m,_) -> encodeFloat m (negate (floatDigits x))
391 scaleFloat k x = case decodeFloat x of
392 (m,n) -> encodeFloat m (n+k)
394 isNaN x = 0 /= isDoubleNaN x
395 isInfinite x = 0 /= isDoubleInfinite x
396 isDenormalized x = 0 /= isDoubleDenormalized x
397 isNegativeZero x = 0 /= isDoubleNegativeZero x
400 instance Show Double where
401 showsPrec x = showSigned showFloat x
402 showList = showList__ (showsPrec 0)
405 %*********************************************************
407 \subsection{@Enum@ instances}
409 %*********************************************************
411 The @Enum@ instances for Floats and Doubles are slightly unusual.
412 The @toEnum@ function truncates numbers to Int. The definitions
413 of @enumFrom@ and @enumFromThen@ allow floats to be used in arithmetic
414 series: [0,0.1 .. 1.0]. However, roundoff errors make these somewhat
415 dubious. This example may have either 10 or 11 elements, depending on
416 how 0.1 is represented.
418 NOTE: The instances for Float and Double do not make use of the default
419 methods for @enumFromTo@ and @enumFromThenTo@, as these rely on there being
420 a `non-lossy' conversion to and from Ints. Instead we make use of the
421 1.2 default methods (back in the days when Enum had Ord as a superclass)
422 for these (@numericEnumFromTo@ and @numericEnumFromThenTo@ below.)
425 instance Enum Float where
429 fromEnum = fromInteger . truncate -- may overflow
430 enumFrom = numericEnumFrom
431 enumFromTo = numericEnumFromTo
432 enumFromThen = numericEnumFromThen
433 enumFromThenTo = numericEnumFromThenTo
435 instance Enum Double where
439 fromEnum = fromInteger . truncate -- may overflow
440 enumFrom = numericEnumFrom
441 enumFromTo = numericEnumFromTo
442 enumFromThen = numericEnumFromThen
443 enumFromThenTo = numericEnumFromThenTo
447 %*********************************************************
449 \subsection{Printing floating point}
451 %*********************************************************
455 showFloat :: (RealFloat a) => a -> ShowS
456 showFloat x = showString (formatRealFloat FFGeneric Nothing x)
458 -- These are the format types. This type is not exported.
460 data FFFormat = FFExponent | FFFixed | FFGeneric
462 formatRealFloat :: (RealFloat a) => FFFormat -> Maybe Int -> a -> String
463 formatRealFloat fmt decs x
465 | isInfinite x = if x < 0 then "-Infinity" else "Infinity"
466 | x < 0 || isNegativeZero x = '-':doFmt fmt (floatToDigits (toInteger base) (-x))
467 | otherwise = doFmt fmt (floatToDigits (toInteger base) x)
471 doFmt format (is, e) =
472 let ds = map intToDigit is in
475 doFmt (if e < 0 || e > 7 then FFExponent else FFFixed)
480 let show_e' = show (e-1) in
483 [d] -> d : ".0e" ++ show_e'
484 (d:ds') -> d : '.' : ds' ++ "e" ++ show_e'
486 let dec' = max dec 1 in
488 [0] -> '0' :'.' : take dec' (repeat '0') ++ "e0"
491 (ei,is') = roundTo base (dec'+1) is
492 (d:ds') = map intToDigit (if ei > 0 then init is' else is')
494 d:'.':ds' ++ 'e':show (e-1+ei)
497 mk0 ls = case ls of { "" -> "0" ; _ -> ls}
502 f 0 s rs = mk0 (reverse s) ++ '.':mk0 rs
503 f n s "" = f (n-1) ('0':s) ""
504 f n s (r:rs) = f (n-1) (r:s) rs
508 let dec' = max dec 0 in
511 (ei,is') = roundTo base (dec' + e) is
512 (ls,rs) = splitAt (e+ei) (map intToDigit is')
514 mk0 ls ++ (if null rs then "" else '.':rs)
517 (ei,is') = roundTo base dec' (replicate (-e) 0 ++ is)
518 d:ds' = map intToDigit (if ei > 0 then is' else 0:is')
523 roundTo :: Int -> Int -> [Int] -> (Int,[Int])
531 f n [] = (0, replicate n 0)
532 f 0 (x:_) = (if x >= b2 then 1 else 0, [])
534 | i' == base = (1,0:ds)
535 | otherwise = (0,i':ds)
541 -- Based on "Printing Floating-Point Numbers Quickly and Accurately"
542 -- by R.G. Burger and R.K. Dybvig in PLDI 96.
543 -- This version uses a much slower logarithm estimator. It should be improved.
545 -- This function returns a list of digits (Ints in [0..base-1]) and an
548 floatToDigits :: (RealFloat a) => Integer -> a -> ([Int], Int)
549 floatToDigits _ 0 = ([0], 0)
550 floatToDigits base x =
552 (f0, e0) = decodeFloat x
553 (minExp0, _) = floatRange x
556 minExp = minExp0 - p -- the real minimum exponent
557 -- Haskell requires that f be adjusted so denormalized numbers
558 -- will have an impossibly low exponent. Adjust for this.
560 let n = minExp - e0 in
561 if n > 0 then (f0 `div` (b^n), e0+n) else (f0, e0)
566 (f*be*b*2, 2*b, be*b, b)
570 if e > minExp && f == b^(p-1) then
571 (f*b*2, b^(-e+1)*2, b, 1)
573 (f*2, b^(-e)*2, 1, 1)
577 if b == 2 && base == 10 then
578 -- logBase 10 2 is slightly bigger than 3/10 so
579 -- the following will err on the low side. Ignoring
580 -- the fraction will make it err even more.
581 -- Haskell promises that p-1 <= logBase b f < p.
582 (p - 1 + e0) * 3 `div` 10
584 ceiling ((log (fromInteger (f+1)) +
585 fromInt e * log (fromInteger b)) /
586 log (fromInteger base))
587 --WAS: fromInt e * log (fromInteger b))
591 if r + mUp <= expt base n * s then n else fixup (n+1)
593 if expt base (-n) * (r + mUp) <= s then n else fixup (n+1)
597 gen ds rn sN mUpN mDnN =
599 (dn, rn') = (rn * base) `divMod` sN
603 case (rn' < mDnN', rn' + mUpN' > sN) of
604 (True, False) -> dn : ds
605 (False, True) -> dn+1 : ds
606 (True, True) -> if rn' * 2 < sN then dn : ds else dn+1 : ds
607 (False, False) -> gen (dn:ds) rn' sN mUpN' mDnN'
611 gen [] r (s * expt base k) mUp mDn
613 let bk = expt base (-k) in
614 gen [] (r * bk) s (mUp * bk) (mDn * bk)
616 (map toInt (reverse rds), k)
621 %*********************************************************
623 \subsection{Converting from a Rational to a RealFloat
625 %*********************************************************
627 [In response to a request for documentation of how fromRational works,
628 Joe Fasel writes:] A quite reasonable request! This code was added to
629 the Prelude just before the 1.2 release, when Lennart, working with an
630 early version of hbi, noticed that (read . show) was not the identity
631 for floating-point numbers. (There was a one-bit error about half the
632 time.) The original version of the conversion function was in fact
633 simply a floating-point divide, as you suggest above. The new version
634 is, I grant you, somewhat denser.
636 Unfortunately, Joe's code doesn't work! Here's an example:
638 main = putStr (shows (1.82173691287639817263897126389712638972163e-300::Double) "\n")
643 1.8217369128763981e-300
648 fromRat :: (RealFloat a) => Rational -> a
652 -- If the exponent of the nearest floating-point number to x
653 -- is e, then the significand is the integer nearest xb^(-e),
654 -- where b is the floating-point radix. We start with a good
655 -- guess for e, and if it is correct, the exponent of the
656 -- floating-point number we construct will again be e. If
657 -- not, one more iteration is needed.
659 f e = if e' == e then y else f e'
660 where y = encodeFloat (round (x * (1 % b)^^e)) e
661 (_,e') = decodeFloat y
664 -- We obtain a trial exponent by doing a floating-point
665 -- division of x's numerator by its denominator. The
666 -- result of this division may not itself be the ultimate
667 -- result, because of an accumulation of three rounding
670 (s,e) = decodeFloat (fromInteger (numerator x) `asTypeOf` x'
671 / fromInteger (denominator x))
674 Now, here's Lennart's code (which works)
677 {-# SPECIALISE fromRat ::
679 Rational -> Float #-}
680 fromRat :: (RealFloat a) => Rational -> a
682 | x == 0 = encodeFloat 0 0 -- Handle exceptional cases
683 | x < 0 = - fromRat' (-x) -- first.
684 | otherwise = fromRat' x
686 -- Conversion process:
687 -- Scale the rational number by the RealFloat base until
688 -- it lies in the range of the mantissa (as used by decodeFloat/encodeFloat).
689 -- Then round the rational to an Integer and encode it with the exponent
690 -- that we got from the scaling.
691 -- To speed up the scaling process we compute the log2 of the number to get
692 -- a first guess of the exponent.
694 fromRat' :: (RealFloat a) => Rational -> a
696 where b = floatRadix r
698 (minExp0, _) = floatRange r
699 minExp = minExp0 - p -- the real minimum exponent
700 xMin = toRational (expt b (p-1))
701 xMax = toRational (expt b p)
702 p0 = (integerLogBase b (numerator x) - integerLogBase b (denominator x) - p) `max` minExp
703 f = if p0 < 0 then 1 % expt b (-p0) else expt b p0 % 1
704 (x', p') = scaleRat (toRational b) minExp xMin xMax p0 (x / f)
705 r = encodeFloat (round x') p'
707 -- Scale x until xMin <= x < xMax, or p (the exponent) <= minExp.
708 scaleRat :: Rational -> Int -> Rational -> Rational -> Int -> Rational -> (Rational, Int)
709 scaleRat b minExp xMin xMax p x
710 | p <= minExp = (x, p)
711 | x >= xMax = scaleRat b minExp xMin xMax (p+1) (x/b)
712 | x < xMin = scaleRat b minExp xMin xMax (p-1) (x*b)
715 -- Exponentiation with a cache for the most common numbers.
716 minExpt, maxExpt :: Int
720 expt :: Integer -> Int -> Integer
722 if base == 2 && n >= minExpt && n <= maxExpt then
727 expts :: Array Int Integer
728 expts = array (minExpt,maxExpt) [(n,2^n) | n <- [minExpt .. maxExpt]]
730 -- Compute the (floor of the) log of i in base b.
731 -- Simplest way would be just divide i by b until it's smaller then b, but that would
732 -- be very slow! We are just slightly more clever.
733 integerLogBase :: Integer -> Integer -> Int
736 | otherwise = doDiv (i `div` (b^l)) l
738 -- Try squaring the base first to cut down the number of divisions.
739 l = 2 * integerLogBase (b*b) i
741 doDiv :: Integer -> Int -> Int
744 | otherwise = doDiv (x `div` b) (y+1)
749 %*********************************************************
751 \subsection{Floating point numeric primops}
753 %*********************************************************
755 Definitions of the boxed PrimOps; these will be
756 used in the case of partial applications, etc.
759 plusFloat, minusFloat, timesFloat, divideFloat :: Float -> Float -> Float
760 plusFloat (F# x) (F# y) = F# (plusFloat# x y)
761 minusFloat (F# x) (F# y) = F# (minusFloat# x y)
762 timesFloat (F# x) (F# y) = F# (timesFloat# x y)
763 divideFloat (F# x) (F# y) = F# (divideFloat# x y)
765 negateFloat :: Float -> Float
766 negateFloat (F# x) = F# (negateFloat# x)
768 gtFloat, geFloat, eqFloat, neFloat, ltFloat, leFloat :: Float -> Float -> Bool
769 gtFloat (F# x) (F# y) = gtFloat# x y
770 geFloat (F# x) (F# y) = geFloat# x y
771 eqFloat (F# x) (F# y) = eqFloat# x y
772 neFloat (F# x) (F# y) = neFloat# x y
773 ltFloat (F# x) (F# y) = ltFloat# x y
774 leFloat (F# x) (F# y) = leFloat# x y
776 float2Int :: Float -> Int
777 float2Int (F# x) = I# (float2Int# x)
779 int2Float :: Int -> Float
780 int2Float (I# x) = F# (int2Float# x)
782 expFloat, logFloat, sqrtFloat :: Float -> Float
783 sinFloat, cosFloat, tanFloat :: Float -> Float
784 asinFloat, acosFloat, atanFloat :: Float -> Float
785 sinhFloat, coshFloat, tanhFloat :: Float -> Float
786 expFloat (F# x) = F# (expFloat# x)
787 logFloat (F# x) = F# (logFloat# x)
788 sqrtFloat (F# x) = F# (sqrtFloat# x)
789 sinFloat (F# x) = F# (sinFloat# x)
790 cosFloat (F# x) = F# (cosFloat# x)
791 tanFloat (F# x) = F# (tanFloat# x)
792 asinFloat (F# x) = F# (asinFloat# x)
793 acosFloat (F# x) = F# (acosFloat# x)
794 atanFloat (F# x) = F# (atanFloat# x)
795 sinhFloat (F# x) = F# (sinhFloat# x)
796 coshFloat (F# x) = F# (coshFloat# x)
797 tanhFloat (F# x) = F# (tanhFloat# x)
799 powerFloat :: Float -> Float -> Float
800 powerFloat (F# x) (F# y) = F# (powerFloat# x y)
802 -- definitions of the boxed PrimOps; these will be
803 -- used in the case of partial applications, etc.
805 plusDouble, minusDouble, timesDouble, divideDouble :: Double -> Double -> Double
806 plusDouble (D# x) (D# y) = D# (x +## y)
807 minusDouble (D# x) (D# y) = D# (x -## y)
808 timesDouble (D# x) (D# y) = D# (x *## y)
809 divideDouble (D# x) (D# y) = D# (x /## y)
811 negateDouble :: Double -> Double
812 negateDouble (D# x) = D# (negateDouble# x)
814 gtDouble, geDouble, eqDouble, neDouble, leDouble, ltDouble :: Double -> Double -> Bool
815 gtDouble (D# x) (D# y) = x >## y
816 geDouble (D# x) (D# y) = x >=## y
817 eqDouble (D# x) (D# y) = x ==## y
818 neDouble (D# x) (D# y) = x /=## y
819 ltDouble (D# x) (D# y) = x <## y
820 leDouble (D# x) (D# y) = x <=## y
822 double2Int :: Double -> Int
823 double2Int (D# x) = I# (double2Int# x)
825 int2Double :: Int -> Double
826 int2Double (I# x) = D# (int2Double# x)
828 double2Float :: Double -> Float
829 double2Float (D# x) = F# (double2Float# x)
831 float2Double :: Float -> Double
832 float2Double (F# x) = D# (float2Double# x)
834 expDouble, logDouble, sqrtDouble :: Double -> Double
835 sinDouble, cosDouble, tanDouble :: Double -> Double
836 asinDouble, acosDouble, atanDouble :: Double -> Double
837 sinhDouble, coshDouble, tanhDouble :: Double -> Double
838 expDouble (D# x) = D# (expDouble# x)
839 logDouble (D# x) = D# (logDouble# x)
840 sqrtDouble (D# x) = D# (sqrtDouble# x)
841 sinDouble (D# x) = D# (sinDouble# x)
842 cosDouble (D# x) = D# (cosDouble# x)
843 tanDouble (D# x) = D# (tanDouble# x)
844 asinDouble (D# x) = D# (asinDouble# x)
845 acosDouble (D# x) = D# (acosDouble# x)
846 atanDouble (D# x) = D# (atanDouble# x)
847 sinhDouble (D# x) = D# (sinhDouble# x)
848 coshDouble (D# x) = D# (coshDouble# x)
849 tanhDouble (D# x) = D# (tanhDouble# x)
851 powerDouble :: Double -> Double -> Double
852 powerDouble (D# x) (D# y) = D# (x **## y)
856 foreign import ccall "__encodeFloat" unsafe
857 encodeFloat# :: Int# -> ByteArray# -> Int -> Float
858 foreign import ccall "__int_encodeFloat" unsafe
859 int_encodeFloat# :: Int# -> Int -> Float
862 foreign import ccall "isFloatNaN" unsafe isFloatNaN :: Float -> Int
863 foreign import ccall "isFloatInfinite" unsafe isFloatInfinite :: Float -> Int
864 foreign import ccall "isFloatDenormalized" unsafe isFloatDenormalized :: Float -> Int
865 foreign import ccall "isFloatNegativeZero" unsafe isFloatNegativeZero :: Float -> Int
868 foreign import ccall "__encodeDouble" unsafe
869 encodeDouble# :: Int# -> ByteArray# -> Int -> Double
870 foreign import ccall "__int_encodeDouble" unsafe
871 int_encodeDouble# :: Int# -> Int -> Double
873 foreign import ccall "isDoubleNaN" unsafe isDoubleNaN :: Double -> Int
874 foreign import ccall "isDoubleInfinite" unsafe isDoubleInfinite :: Double -> Int
875 foreign import ccall "isDoubleDenormalized" unsafe isDoubleDenormalized :: Double -> Int
876 foreign import ccall "isDoubleNegativeZero" unsafe isDoubleNegativeZero :: Double -> Int