2 {-# OPTIONS_GHC -fno-implicit-prelude #-}
3 -----------------------------------------------------------------------------
6 -- Copyright : (c) The University of Glasgow 1994-2002
7 -- License : see libraries/base/LICENSE
9 -- Maintainer : cvs-ghc@haskell.org
10 -- Stability : internal
11 -- Portability : non-portable (GHC Extensions)
13 -- The types 'Float' and 'Double', and the classes 'Floating' and 'RealFloat'.
15 -----------------------------------------------------------------------------
17 #include "ieee-flpt.h"
20 module GHC.Float( module GHC.Float, Float#, Double# ) where
35 %*********************************************************
37 \subsection{Standard numeric classes}
39 %*********************************************************
42 -- | Trigonometric and hyperbolic functions and related functions.
44 -- Minimal complete definition:
45 -- 'pi', 'exp', 'log', 'sin', 'cos', 'sinh', 'cosh'
46 -- 'asin', 'acos', 'atan', 'asinh', 'acosh' and 'atanh'
47 class (Fractional a) => Floating a where
49 exp, log, sqrt :: a -> a
50 (**), logBase :: a -> a -> a
51 sin, cos, tan :: a -> a
52 asin, acos, atan :: a -> a
53 sinh, cosh, tanh :: a -> a
54 asinh, acosh, atanh :: a -> a
56 x ** y = exp (log x * y)
57 logBase x y = log y / log x
60 tanh x = sinh x / cosh x
62 -- | Efficient, machine-independent access to the components of a
63 -- floating-point number.
65 -- Minimal complete definition:
66 -- all except 'exponent', 'significand', 'scaleFloat' and 'atan2'
67 class (RealFrac a, Floating a) => RealFloat a where
68 -- | a constant function, returning the radix of the representation
70 floatRadix :: a -> Integer
71 -- | a constant function, returning the number of digits of
72 -- 'floatRadix' in the significand
73 floatDigits :: a -> Int
74 -- | a constant function, returning the lowest and highest values
75 -- the exponent may assume
76 floatRange :: a -> (Int,Int)
77 -- | The function 'decodeFloat' applied to a real floating-point
78 -- number returns the significand expressed as an 'Integer' and an
79 -- appropriately scaled exponent (an 'Int'). If @'decodeFloat' x@
80 -- yields @(m,n)@, then @x@ is equal in value to @m*b^^n@, where @b@
81 -- is the floating-point radix, and furthermore, either @m@ and @n@
82 -- are both zero or else @b^(d-1) <= m < b^d@, where @d@ is the value
83 -- of @'floatDigits' x@. In particular, @'decodeFloat' 0 = (0,0)@.
84 decodeFloat :: a -> (Integer,Int)
85 -- | 'encodeFloat' performs the inverse of 'decodeFloat'
86 encodeFloat :: Integer -> Int -> a
87 -- | the second component of 'decodeFloat'.
89 -- | the first component of 'decodeFloat', scaled to lie in the open
90 -- interval (@-1@,@1@)
92 -- | multiplies a floating-point number by an integer power of the radix
93 scaleFloat :: Int -> a -> a
94 -- | 'True' if the argument is an IEEE \"not-a-number\" (NaN) value
96 -- | 'True' if the argument is an IEEE infinity or negative infinity
97 isInfinite :: a -> Bool
98 -- | 'True' if the argument is too small to be represented in
100 isDenormalized :: a -> Bool
101 -- | 'True' if the argument is an IEEE negative zero
102 isNegativeZero :: a -> Bool
103 -- | 'True' if the argument is an IEEE floating point number
105 -- | a version of arctangent taking two real floating-point arguments.
106 -- For real floating @x@ and @y@, @'atan2' y x@ computes the angle
107 -- (from the positive x-axis) of the vector from the origin to the
108 -- point @(x,y)@. @'atan2' y x@ returns a value in the range [@-pi@,
109 -- @pi@]. It follows the Common Lisp semantics for the origin when
110 -- signed zeroes are supported. @'atan2' y 1@, with @y@ in a type
111 -- that is 'RealFloat', should return the same value as @'atan' y@.
112 -- A default definition of 'atan2' is provided, but implementors
113 -- can provide a more accurate implementation.
117 exponent x = if m == 0 then 0 else n + floatDigits x
118 where (m,n) = decodeFloat x
120 significand x = encodeFloat m (negate (floatDigits x))
121 where (m,_) = decodeFloat x
123 scaleFloat k x = encodeFloat m (n+k)
124 where (m,n) = decodeFloat x
128 | x == 0 && y > 0 = pi/2
129 | x < 0 && y > 0 = pi + atan (y/x)
130 |(x <= 0 && y < 0) ||
131 (x < 0 && isNegativeZero y) ||
132 (isNegativeZero x && isNegativeZero y)
134 | y == 0 && (x < 0 || isNegativeZero x)
135 = pi -- must be after the previous test on zero y
136 | x==0 && y==0 = y -- must be after the other double zero tests
137 | otherwise = x + y -- x or y is a NaN, return a NaN (via +)
141 %*********************************************************
143 \subsection{Type @Integer@, @Float@, @Double@}
145 %*********************************************************
148 -- | Single-precision floating point numbers.
149 -- It is desirable that this type be at least equal in range and precision
150 -- to the IEEE single-precision type.
151 data Float = F# Float#
153 -- | Double-precision floating point numbers.
154 -- It is desirable that this type be at least equal in range and precision
155 -- to the IEEE double-precision type.
156 data Double = D# Double#
160 %*********************************************************
162 \subsection{Type @Float@}
164 %*********************************************************
167 instance Eq Float where
168 (F# x) == (F# y) = x `eqFloat#` y
170 instance Ord Float where
171 (F# x) `compare` (F# y) | x `ltFloat#` y = LT
172 | x `eqFloat#` y = EQ
175 (F# x) < (F# y) = x `ltFloat#` y
176 (F# x) <= (F# y) = x `leFloat#` y
177 (F# x) >= (F# y) = x `geFloat#` y
178 (F# x) > (F# y) = x `gtFloat#` y
180 instance Num Float where
181 (+) x y = plusFloat x y
182 (-) x y = minusFloat x y
183 negate x = negateFloat x
184 (*) x y = timesFloat x y
186 | otherwise = negateFloat x
187 signum x | x == 0.0 = 0
189 | otherwise = negate 1
191 {-# INLINE fromInteger #-}
192 fromInteger n = encodeFloat n 0
193 -- It's important that encodeFloat inlines here, and that
194 -- fromInteger in turn inlines,
195 -- so that if fromInteger is applied to an (S# i) the right thing happens
197 instance Real Float where
198 toRational x = (m%1)*(b%1)^^n
199 where (m,n) = decodeFloat x
202 instance Fractional Float where
203 (/) x y = divideFloat x y
204 fromRational x = fromRat x
207 {-# RULES "truncate/Float->Int" truncate = float2Int #-}
208 instance RealFrac Float where
210 {-# SPECIALIZE properFraction :: Float -> (Int, Float) #-}
211 {-# SPECIALIZE round :: Float -> Int #-}
213 {-# SPECIALIZE properFraction :: Float -> (Integer, Float) #-}
214 {-# SPECIALIZE round :: Float -> Integer #-}
216 -- ceiling, floor, and truncate are all small
217 {-# INLINE ceiling #-}
219 {-# INLINE truncate #-}
222 = case (decodeFloat x) of { (m,n) ->
223 let b = floatRadix x in
225 (fromInteger m * fromInteger b ^ n, 0.0)
227 case (quotRem m (b^(negate n))) of { (w,r) ->
228 (fromInteger w, encodeFloat r n)
232 truncate x = case properFraction x of
235 round x = case properFraction x of
237 m = if r < 0.0 then n - 1 else n + 1
238 half_down = abs r - 0.5
240 case (compare half_down 0.0) of
242 EQ -> if even n then n else m
245 ceiling x = case properFraction x of
246 (n,r) -> if r > 0.0 then n + 1 else n
248 floor x = case properFraction x of
249 (n,r) -> if r < 0.0 then n - 1 else n
251 instance Floating Float where
252 pi = 3.141592653589793238
265 (**) x y = powerFloat x y
266 logBase x y = log y / log x
268 asinh x = log (x + sqrt (1.0+x*x))
269 acosh x = log (x + (x+1.0) * sqrt ((x-1.0)/(x+1.0)))
270 atanh x = log ((x+1.0) / sqrt (1.0-x*x))
272 instance RealFloat Float where
273 floatRadix _ = FLT_RADIX -- from float.h
274 floatDigits _ = FLT_MANT_DIG -- ditto
275 floatRange _ = (FLT_MIN_EXP, FLT_MAX_EXP) -- ditto
278 = case decodeFloat# f# of
279 (# exp#, s#, d# #) -> (J# s# d#, I# exp#)
281 encodeFloat (S# i) j = int_encodeFloat# i j
282 encodeFloat (J# s# d#) e = encodeFloat# s# d# e
284 exponent x = case decodeFloat x of
285 (m,n) -> if m == 0 then 0 else n + floatDigits x
287 significand x = case decodeFloat x of
288 (m,_) -> encodeFloat m (negate (floatDigits x))
290 scaleFloat k x = case decodeFloat x of
291 (m,n) -> encodeFloat m (n+k)
292 isNaN x = 0 /= isFloatNaN x
293 isInfinite x = 0 /= isFloatInfinite x
294 isDenormalized x = 0 /= isFloatDenormalized x
295 isNegativeZero x = 0 /= isFloatNegativeZero x
298 instance Show Float where
299 showsPrec x = showSigned showFloat x
300 showList = showList__ (showsPrec 0)
303 %*********************************************************
305 \subsection{Type @Double@}
307 %*********************************************************
310 instance Eq Double where
311 (D# x) == (D# y) = x ==## y
313 instance Ord Double where
314 (D# x) `compare` (D# y) | x <## y = LT
318 (D# x) < (D# y) = x <## y
319 (D# x) <= (D# y) = x <=## y
320 (D# x) >= (D# y) = x >=## y
321 (D# x) > (D# y) = x >## y
323 instance Num Double where
324 (+) x y = plusDouble x y
325 (-) x y = minusDouble x y
326 negate x = negateDouble x
327 (*) x y = timesDouble x y
329 | otherwise = negateDouble x
330 signum x | x == 0.0 = 0
332 | otherwise = negate 1
334 {-# INLINE fromInteger #-}
335 -- See comments with Num Float
336 fromInteger (S# i#) = case (int2Double# i#) of { d# -> D# d# }
337 fromInteger (J# s# d#) = encodeDouble# s# d# 0
340 instance Real Double where
341 toRational x = (m%1)*(b%1)^^n
342 where (m,n) = decodeFloat x
345 instance Fractional Double where
346 (/) x y = divideDouble x y
347 fromRational x = fromRat x
350 instance Floating Double where
351 pi = 3.141592653589793238
354 sqrt x = sqrtDouble x
358 asin x = asinDouble x
359 acos x = acosDouble x
360 atan x = atanDouble x
361 sinh x = sinhDouble x
362 cosh x = coshDouble x
363 tanh x = tanhDouble x
364 (**) x y = powerDouble x y
365 logBase x y = log y / log x
367 asinh x = log (x + sqrt (1.0+x*x))
368 acosh x = log (x + (x+1.0) * sqrt ((x-1.0)/(x+1.0)))
369 atanh x = log ((x+1.0) / sqrt (1.0-x*x))
371 {-# RULES "truncate/Double->Int" truncate = double2Int #-}
372 instance RealFrac Double where
374 {-# SPECIALIZE properFraction :: Double -> (Int, Double) #-}
375 {-# SPECIALIZE round :: Double -> Int #-}
377 {-# SPECIALIZE properFraction :: Double -> (Integer, Double) #-}
378 {-# SPECIALIZE round :: Double -> Integer #-}
380 -- ceiling, floor, and truncate are all small
381 {-# INLINE ceiling #-}
383 {-# INLINE truncate #-}
386 = case (decodeFloat x) of { (m,n) ->
387 let b = floatRadix x in
389 (fromInteger m * fromInteger b ^ n, 0.0)
391 case (quotRem m (b^(negate n))) of { (w,r) ->
392 (fromInteger w, encodeFloat r n)
396 truncate x = case properFraction x of
399 round x = case properFraction x of
401 m = if r < 0.0 then n - 1 else n + 1
402 half_down = abs r - 0.5
404 case (compare half_down 0.0) of
406 EQ -> if even n then n else m
409 ceiling x = case properFraction x of
410 (n,r) -> if r > 0.0 then n + 1 else n
412 floor x = case properFraction x of
413 (n,r) -> if r < 0.0 then n - 1 else n
415 instance RealFloat Double where
416 floatRadix _ = FLT_RADIX -- from float.h
417 floatDigits _ = DBL_MANT_DIG -- ditto
418 floatRange _ = (DBL_MIN_EXP, DBL_MAX_EXP) -- ditto
421 = case decodeDouble# x# of
422 (# exp#, s#, d# #) -> (J# s# d#, I# exp#)
424 encodeFloat (S# i) j = int_encodeDouble# i j
425 encodeFloat (J# s# d#) e = encodeDouble# s# d# e
427 exponent x = case decodeFloat x of
428 (m,n) -> if m == 0 then 0 else n + floatDigits x
430 significand x = case decodeFloat x of
431 (m,_) -> encodeFloat m (negate (floatDigits x))
433 scaleFloat k x = case decodeFloat x of
434 (m,n) -> encodeFloat m (n+k)
436 isNaN x = 0 /= isDoubleNaN x
437 isInfinite x = 0 /= isDoubleInfinite x
438 isDenormalized x = 0 /= isDoubleDenormalized x
439 isNegativeZero x = 0 /= isDoubleNegativeZero x
442 instance Show Double where
443 showsPrec x = showSigned showFloat x
444 showList = showList__ (showsPrec 0)
447 %*********************************************************
449 \subsection{@Enum@ instances}
451 %*********************************************************
453 The @Enum@ instances for Floats and Doubles are slightly unusual.
454 The @toEnum@ function truncates numbers to Int. The definitions
455 of @enumFrom@ and @enumFromThen@ allow floats to be used in arithmetic
456 series: [0,0.1 .. 1.0]. However, roundoff errors make these somewhat
457 dubious. This example may have either 10 or 11 elements, depending on
458 how 0.1 is represented.
460 NOTE: The instances for Float and Double do not make use of the default
461 methods for @enumFromTo@ and @enumFromThenTo@, as these rely on there being
462 a `non-lossy' conversion to and from Ints. Instead we make use of the
463 1.2 default methods (back in the days when Enum had Ord as a superclass)
464 for these (@numericEnumFromTo@ and @numericEnumFromThenTo@ below.)
467 instance Enum Float where
471 fromEnum = fromInteger . truncate -- may overflow
472 enumFrom = numericEnumFrom
473 enumFromTo = numericEnumFromTo
474 enumFromThen = numericEnumFromThen
475 enumFromThenTo = numericEnumFromThenTo
477 instance Enum Double where
481 fromEnum = fromInteger . truncate -- may overflow
482 enumFrom = numericEnumFrom
483 enumFromTo = numericEnumFromTo
484 enumFromThen = numericEnumFromThen
485 enumFromThenTo = numericEnumFromThenTo
489 %*********************************************************
491 \subsection{Printing floating point}
493 %*********************************************************
497 -- | Show a signed 'RealFloat' value to full precision
498 -- using standard decimal notation for arguments whose absolute value lies
499 -- between @0.1@ and @9,999,999@, and scientific notation otherwise.
500 showFloat :: (RealFloat a) => a -> ShowS
501 showFloat x = showString (formatRealFloat FFGeneric Nothing x)
503 -- These are the format types. This type is not exported.
505 data FFFormat = FFExponent | FFFixed | FFGeneric
507 formatRealFloat :: (RealFloat a) => FFFormat -> Maybe Int -> a -> String
508 formatRealFloat fmt decs x
510 | isInfinite x = if x < 0 then "-Infinity" else "Infinity"
511 | x < 0 || isNegativeZero x = '-':doFmt fmt (floatToDigits (toInteger base) (-x))
512 | otherwise = doFmt fmt (floatToDigits (toInteger base) x)
516 doFmt format (is, e) =
517 let ds = map intToDigit is in
520 doFmt (if e < 0 || e > 7 then FFExponent else FFFixed)
525 let show_e' = show (e-1) in
528 [d] -> d : ".0e" ++ show_e'
529 (d:ds') -> d : '.' : ds' ++ "e" ++ show_e'
531 let dec' = max dec 1 in
533 [0] -> '0' :'.' : take dec' (repeat '0') ++ "e0"
536 (ei,is') = roundTo base (dec'+1) is
537 (d:ds') = map intToDigit (if ei > 0 then init is' else is')
539 d:'.':ds' ++ 'e':show (e-1+ei)
542 mk0 ls = case ls of { "" -> "0" ; _ -> ls}
546 | e <= 0 -> "0." ++ replicate (-e) '0' ++ ds
549 f 0 s rs = mk0 (reverse s) ++ '.':mk0 rs
550 f n s "" = f (n-1) ('0':s) ""
551 f n s (r:rs) = f (n-1) (r:s) rs
555 let dec' = max dec 0 in
558 (ei,is') = roundTo base (dec' + e) is
559 (ls,rs) = splitAt (e+ei) (map intToDigit is')
561 mk0 ls ++ (if null rs then "" else '.':rs)
564 (ei,is') = roundTo base dec' (replicate (-e) 0 ++ is)
565 d:ds' = map intToDigit (if ei > 0 then is' else 0:is')
567 d : (if null ds' then "" else '.':ds')
570 roundTo :: Int -> Int -> [Int] -> (Int,[Int])
578 f n [] = (0, replicate n 0)
579 f 0 (x:_) = (if x >= b2 then 1 else 0, [])
581 | i' == base = (1,0:ds)
582 | otherwise = (0,i':ds)
587 -- Based on "Printing Floating-Point Numbers Quickly and Accurately"
588 -- by R.G. Burger and R.K. Dybvig in PLDI 96.
589 -- This version uses a much slower logarithm estimator. It should be improved.
591 -- | 'floatToDigits' takes a base and a non-negative 'RealFloat' number,
592 -- and returns a list of digits and an exponent.
593 -- In particular, if @x>=0@, and
595 -- > floatToDigits base x = ([d1,d2,...,dn], e)
601 -- (2) @x = 0.d1d2...dn * (base**e)@
603 -- (3) @0 <= di <= base-1@
605 floatToDigits :: (RealFloat a) => Integer -> a -> ([Int], Int)
606 floatToDigits _ 0 = ([0], 0)
607 floatToDigits base x =
609 (f0, e0) = decodeFloat x
610 (minExp0, _) = floatRange x
613 minExp = minExp0 - p -- the real minimum exponent
614 -- Haskell requires that f be adjusted so denormalized numbers
615 -- will have an impossibly low exponent. Adjust for this.
617 let n = minExp - e0 in
618 if n > 0 then (f0 `div` (b^n), e0+n) else (f0, e0)
623 (f*be*b*2, 2*b, be*b, b)
627 if e > minExp && f == b^(p-1) then
628 (f*b*2, b^(-e+1)*2, b, 1)
630 (f*2, b^(-e)*2, 1, 1)
634 if b == 2 && base == 10 then
635 -- logBase 10 2 is slightly bigger than 3/10 so
636 -- the following will err on the low side. Ignoring
637 -- the fraction will make it err even more.
638 -- Haskell promises that p-1 <= logBase b f < p.
639 (p - 1 + e0) * 3 `div` 10
641 ceiling ((log (fromInteger (f+1)) +
642 fromInteger (int2Integer e) * log (fromInteger b)) /
643 log (fromInteger base))
644 --WAS: fromInt e * log (fromInteger b))
648 if r + mUp <= expt base n * s then n else fixup (n+1)
650 if expt base (-n) * (r + mUp) <= s then n else fixup (n+1)
654 gen ds rn sN mUpN mDnN =
656 (dn, rn') = (rn * base) `divMod` sN
660 case (rn' < mDnN', rn' + mUpN' > sN) of
661 (True, False) -> dn : ds
662 (False, True) -> dn+1 : ds
663 (True, True) -> if rn' * 2 < sN then dn : ds else dn+1 : ds
664 (False, False) -> gen (dn:ds) rn' sN mUpN' mDnN'
668 gen [] r (s * expt base k) mUp mDn
670 let bk = expt base (-k) in
671 gen [] (r * bk) s (mUp * bk) (mDn * bk)
673 (map fromIntegral (reverse rds), k)
678 %*********************************************************
680 \subsection{Converting from a Rational to a RealFloat
682 %*********************************************************
684 [In response to a request for documentation of how fromRational works,
685 Joe Fasel writes:] A quite reasonable request! This code was added to
686 the Prelude just before the 1.2 release, when Lennart, working with an
687 early version of hbi, noticed that (read . show) was not the identity
688 for floating-point numbers. (There was a one-bit error about half the
689 time.) The original version of the conversion function was in fact
690 simply a floating-point divide, as you suggest above. The new version
691 is, I grant you, somewhat denser.
693 Unfortunately, Joe's code doesn't work! Here's an example:
695 main = putStr (shows (1.82173691287639817263897126389712638972163e-300::Double) "\n")
700 1.8217369128763981e-300
705 fromRat :: (RealFloat a) => Rational -> a
709 -- If the exponent of the nearest floating-point number to x
710 -- is e, then the significand is the integer nearest xb^(-e),
711 -- where b is the floating-point radix. We start with a good
712 -- guess for e, and if it is correct, the exponent of the
713 -- floating-point number we construct will again be e. If
714 -- not, one more iteration is needed.
716 f e = if e' == e then y else f e'
717 where y = encodeFloat (round (x * (1 % b)^^e)) e
718 (_,e') = decodeFloat y
721 -- We obtain a trial exponent by doing a floating-point
722 -- division of x's numerator by its denominator. The
723 -- result of this division may not itself be the ultimate
724 -- result, because of an accumulation of three rounding
727 (s,e) = decodeFloat (fromInteger (numerator x) `asTypeOf` x'
728 / fromInteger (denominator x))
731 Now, here's Lennart's code (which works)
734 -- | Converts a 'Rational' value into any type in class 'RealFloat'.
735 {-# SPECIALISE fromRat :: Rational -> Double,
736 Rational -> Float #-}
737 fromRat :: (RealFloat a) => Rational -> a
739 -- Deal with special cases first, delegating the real work to fromRat'
740 fromRat (n :% 0) | n > 0 = 1/0 -- +Infinity
741 | n == 0 = 0/0 -- NaN
742 | n < 0 = -1/0 -- -Infinity
744 fromRat (n :% d) | n > 0 = fromRat' (n :% d)
745 | n == 0 = encodeFloat 0 0 -- Zero
746 | n < 0 = - fromRat' ((-n) :% d)
748 -- Conversion process:
749 -- Scale the rational number by the RealFloat base until
750 -- it lies in the range of the mantissa (as used by decodeFloat/encodeFloat).
751 -- Then round the rational to an Integer and encode it with the exponent
752 -- that we got from the scaling.
753 -- To speed up the scaling process we compute the log2 of the number to get
754 -- a first guess of the exponent.
756 fromRat' :: (RealFloat a) => Rational -> a
757 -- Invariant: argument is strictly positive
759 where b = floatRadix r
761 (minExp0, _) = floatRange r
762 minExp = minExp0 - p -- the real minimum exponent
763 xMin = toRational (expt b (p-1))
764 xMax = toRational (expt b p)
765 p0 = (integerLogBase b (numerator x) - integerLogBase b (denominator x) - p) `max` minExp
766 f = if p0 < 0 then 1 % expt b (-p0) else expt b p0 % 1
767 (x', p') = scaleRat (toRational b) minExp xMin xMax p0 (x / f)
768 r = encodeFloat (round x') p'
770 -- Scale x until xMin <= x < xMax, or p (the exponent) <= minExp.
771 scaleRat :: Rational -> Int -> Rational -> Rational -> Int -> Rational -> (Rational, Int)
772 scaleRat b minExp xMin xMax p x
773 | p <= minExp = (x, p)
774 | x >= xMax = scaleRat b minExp xMin xMax (p+1) (x/b)
775 | x < xMin = scaleRat b minExp xMin xMax (p-1) (x*b)
778 -- Exponentiation with a cache for the most common numbers.
779 minExpt, maxExpt :: Int
783 expt :: Integer -> Int -> Integer
785 if base == 2 && n >= minExpt && n <= maxExpt then
790 expts :: Array Int Integer
791 expts = array (minExpt,maxExpt) [(n,2^n) | n <- [minExpt .. maxExpt]]
793 -- Compute the (floor of the) log of i in base b.
794 -- Simplest way would be just divide i by b until it's smaller then b, but that would
795 -- be very slow! We are just slightly more clever.
796 integerLogBase :: Integer -> Integer -> Int
799 | otherwise = doDiv (i `div` (b^l)) l
801 -- Try squaring the base first to cut down the number of divisions.
802 l = 2 * integerLogBase (b*b) i
804 doDiv :: Integer -> Int -> Int
807 | otherwise = doDiv (x `div` b) (y+1)
812 %*********************************************************
814 \subsection{Floating point numeric primops}
816 %*********************************************************
818 Definitions of the boxed PrimOps; these will be
819 used in the case of partial applications, etc.
822 plusFloat, minusFloat, timesFloat, divideFloat :: Float -> Float -> Float
823 plusFloat (F# x) (F# y) = F# (plusFloat# x y)
824 minusFloat (F# x) (F# y) = F# (minusFloat# x y)
825 timesFloat (F# x) (F# y) = F# (timesFloat# x y)
826 divideFloat (F# x) (F# y) = F# (divideFloat# x y)
828 negateFloat :: Float -> Float
829 negateFloat (F# x) = F# (negateFloat# x)
831 gtFloat, geFloat, eqFloat, neFloat, ltFloat, leFloat :: Float -> Float -> Bool
832 gtFloat (F# x) (F# y) = gtFloat# x y
833 geFloat (F# x) (F# y) = geFloat# x y
834 eqFloat (F# x) (F# y) = eqFloat# x y
835 neFloat (F# x) (F# y) = neFloat# x y
836 ltFloat (F# x) (F# y) = ltFloat# x y
837 leFloat (F# x) (F# y) = leFloat# x y
839 float2Int :: Float -> Int
840 float2Int (F# x) = I# (float2Int# x)
842 int2Float :: Int -> Float
843 int2Float (I# x) = F# (int2Float# x)
845 expFloat, logFloat, sqrtFloat :: Float -> Float
846 sinFloat, cosFloat, tanFloat :: Float -> Float
847 asinFloat, acosFloat, atanFloat :: Float -> Float
848 sinhFloat, coshFloat, tanhFloat :: Float -> Float
849 expFloat (F# x) = F# (expFloat# x)
850 logFloat (F# x) = F# (logFloat# x)
851 sqrtFloat (F# x) = F# (sqrtFloat# x)
852 sinFloat (F# x) = F# (sinFloat# x)
853 cosFloat (F# x) = F# (cosFloat# x)
854 tanFloat (F# x) = F# (tanFloat# x)
855 asinFloat (F# x) = F# (asinFloat# x)
856 acosFloat (F# x) = F# (acosFloat# x)
857 atanFloat (F# x) = F# (atanFloat# x)
858 sinhFloat (F# x) = F# (sinhFloat# x)
859 coshFloat (F# x) = F# (coshFloat# x)
860 tanhFloat (F# x) = F# (tanhFloat# x)
862 powerFloat :: Float -> Float -> Float
863 powerFloat (F# x) (F# y) = F# (powerFloat# x y)
865 -- definitions of the boxed PrimOps; these will be
866 -- used in the case of partial applications, etc.
868 plusDouble, minusDouble, timesDouble, divideDouble :: Double -> Double -> Double
869 plusDouble (D# x) (D# y) = D# (x +## y)
870 minusDouble (D# x) (D# y) = D# (x -## y)
871 timesDouble (D# x) (D# y) = D# (x *## y)
872 divideDouble (D# x) (D# y) = D# (x /## y)
874 negateDouble :: Double -> Double
875 negateDouble (D# x) = D# (negateDouble# x)
877 gtDouble, geDouble, eqDouble, neDouble, leDouble, ltDouble :: Double -> Double -> Bool
878 gtDouble (D# x) (D# y) = x >## y
879 geDouble (D# x) (D# y) = x >=## y
880 eqDouble (D# x) (D# y) = x ==## y
881 neDouble (D# x) (D# y) = x /=## y
882 ltDouble (D# x) (D# y) = x <## y
883 leDouble (D# x) (D# y) = x <=## y
885 double2Int :: Double -> Int
886 double2Int (D# x) = I# (double2Int# x)
888 int2Double :: Int -> Double
889 int2Double (I# x) = D# (int2Double# x)
891 double2Float :: Double -> Float
892 double2Float (D# x) = F# (double2Float# x)
894 float2Double :: Float -> Double
895 float2Double (F# x) = D# (float2Double# x)
897 expDouble, logDouble, sqrtDouble :: Double -> Double
898 sinDouble, cosDouble, tanDouble :: Double -> Double
899 asinDouble, acosDouble, atanDouble :: Double -> Double
900 sinhDouble, coshDouble, tanhDouble :: Double -> Double
901 expDouble (D# x) = D# (expDouble# x)
902 logDouble (D# x) = D# (logDouble# x)
903 sqrtDouble (D# x) = D# (sqrtDouble# x)
904 sinDouble (D# x) = D# (sinDouble# x)
905 cosDouble (D# x) = D# (cosDouble# x)
906 tanDouble (D# x) = D# (tanDouble# x)
907 asinDouble (D# x) = D# (asinDouble# x)
908 acosDouble (D# x) = D# (acosDouble# x)
909 atanDouble (D# x) = D# (atanDouble# x)
910 sinhDouble (D# x) = D# (sinhDouble# x)
911 coshDouble (D# x) = D# (coshDouble# x)
912 tanhDouble (D# x) = D# (tanhDouble# x)
914 powerDouble :: Double -> Double -> Double
915 powerDouble (D# x) (D# y) = D# (x **## y)
919 foreign import ccall unsafe "__encodeFloat"
920 encodeFloat# :: Int# -> ByteArray# -> Int -> Float
921 foreign import ccall unsafe "__int_encodeFloat"
922 int_encodeFloat# :: Int# -> Int -> Float
925 foreign import ccall unsafe "isFloatNaN" isFloatNaN :: Float -> Int
926 foreign import ccall unsafe "isFloatInfinite" isFloatInfinite :: Float -> Int
927 foreign import ccall unsafe "isFloatDenormalized" isFloatDenormalized :: Float -> Int
928 foreign import ccall unsafe "isFloatNegativeZero" isFloatNegativeZero :: Float -> Int
931 foreign import ccall unsafe "__encodeDouble"
932 encodeDouble# :: Int# -> ByteArray# -> Int -> Double
933 foreign import ccall unsafe "__int_encodeDouble"
934 int_encodeDouble# :: Int# -> Int -> Double
936 foreign import ccall unsafe "isDoubleNaN" isDoubleNaN :: Double -> Int
937 foreign import ccall unsafe "isDoubleInfinite" isDoubleInfinite :: Double -> Int
938 foreign import ccall unsafe "isDoubleDenormalized" isDoubleDenormalized :: Double -> Int
939 foreign import ccall unsafe "isDoubleNegativeZero" isDoubleNegativeZero :: Double -> Int
942 %*********************************************************
944 \subsection{Coercion rules}
946 %*********************************************************
950 "fromIntegral/Int->Float" fromIntegral = int2Float
951 "fromIntegral/Int->Double" fromIntegral = int2Double
952 "realToFrac/Float->Float" realToFrac = id :: Float -> Float
953 "realToFrac/Float->Double" realToFrac = float2Double
954 "realToFrac/Double->Float" realToFrac = double2Float
955 "realToFrac/Double->Double" realToFrac = id :: Double -> Double