2 {-# OPTIONS_GHC -XNoImplicitPrelude #-}
3 -- We believe we could deorphan this module, by moving lots of things
4 -- around, but we haven't got there yet:
5 {-# OPTIONS_GHC -fno-warn-orphans #-}
6 {-# OPTIONS_HADDOCK hide #-}
7 -----------------------------------------------------------------------------
10 -- Copyright : (c) The University of Glasgow 1994-2002
11 -- License : see libraries/base/LICENSE
13 -- Maintainer : cvs-ghc@haskell.org
14 -- Stability : internal
15 -- Portability : non-portable (GHC Extensions)
17 -- The types 'Float' and 'Double', and the classes 'Floating' and 'RealFloat'.
19 -----------------------------------------------------------------------------
21 #include "ieee-flpt.h"
24 module GHC.Float( module GHC.Float, Float(..), Double(..), Float#, Double# )
41 %*********************************************************
43 \subsection{Standard numeric classes}
45 %*********************************************************
48 -- | Trigonometric and hyperbolic functions and related functions.
50 -- Minimal complete definition:
51 -- 'pi', 'exp', 'log', 'sin', 'cos', 'sinh', 'cosh',
52 -- 'asin', 'acos', 'atan', 'asinh', 'acosh' and 'atanh'
53 class (Fractional a) => Floating a where
55 exp, log, sqrt :: a -> a
56 (**), logBase :: a -> a -> a
57 sin, cos, tan :: a -> a
58 asin, acos, atan :: a -> a
59 sinh, cosh, tanh :: a -> a
60 asinh, acosh, atanh :: a -> a
63 {-# INLINE logBase #-}
67 x ** y = exp (log x * y)
68 logBase x y = log y / log x
71 tanh x = sinh x / cosh x
73 -- | Efficient, machine-independent access to the components of a
74 -- floating-point number.
76 -- Minimal complete definition:
77 -- all except 'exponent', 'significand', 'scaleFloat' and 'atan2'
78 class (RealFrac a, Floating a) => RealFloat a where
79 -- | a constant function, returning the radix of the representation
81 floatRadix :: a -> Integer
82 -- | a constant function, returning the number of digits of
83 -- 'floatRadix' in the significand
84 floatDigits :: a -> Int
85 -- | a constant function, returning the lowest and highest values
86 -- the exponent may assume
87 floatRange :: a -> (Int,Int)
88 -- | The function 'decodeFloat' applied to a real floating-point
89 -- number returns the significand expressed as an 'Integer' and an
90 -- appropriately scaled exponent (an 'Int'). If @'decodeFloat' x@
91 -- yields @(m,n)@, then @x@ is equal in value to @m*b^^n@, where @b@
92 -- is the floating-point radix, and furthermore, either @m@ and @n@
93 -- are both zero or else @b^(d-1) <= m < b^d@, where @d@ is the value
94 -- of @'floatDigits' x@. In particular, @'decodeFloat' 0 = (0,0)@.
95 decodeFloat :: a -> (Integer,Int)
96 -- | 'encodeFloat' performs the inverse of 'decodeFloat'
97 encodeFloat :: Integer -> Int -> a
98 -- | the second component of 'decodeFloat'.
100 -- | the first component of 'decodeFloat', scaled to lie in the open
101 -- interval (@-1@,@1@)
102 significand :: a -> a
103 -- | multiplies a floating-point number by an integer power of the radix
104 scaleFloat :: Int -> a -> a
105 -- | 'True' if the argument is an IEEE \"not-a-number\" (NaN) value
107 -- | 'True' if the argument is an IEEE infinity or negative infinity
108 isInfinite :: a -> Bool
109 -- | 'True' if the argument is too small to be represented in
111 isDenormalized :: a -> Bool
112 -- | 'True' if the argument is an IEEE negative zero
113 isNegativeZero :: a -> Bool
114 -- | 'True' if the argument is an IEEE floating point number
116 -- | a version of arctangent taking two real floating-point arguments.
117 -- For real floating @x@ and @y@, @'atan2' y x@ computes the angle
118 -- (from the positive x-axis) of the vector from the origin to the
119 -- point @(x,y)@. @'atan2' y x@ returns a value in the range [@-pi@,
120 -- @pi@]. It follows the Common Lisp semantics for the origin when
121 -- signed zeroes are supported. @'atan2' y 1@, with @y@ in a type
122 -- that is 'RealFloat', should return the same value as @'atan' y@.
123 -- A default definition of 'atan2' is provided, but implementors
124 -- can provide a more accurate implementation.
128 exponent x = if m == 0 then 0 else n + floatDigits x
129 where (m,n) = decodeFloat x
131 significand x = encodeFloat m (negate (floatDigits x))
132 where (m,_) = decodeFloat x
134 scaleFloat k x = encodeFloat m (n + clamp b k)
135 where (m,n) = decodeFloat x
139 -- n+k may overflow, which would lead
140 -- to wrong results, hence we clamp the
141 -- scaling parameter.
142 -- If n + k would be larger than h,
143 -- n + clamp b k must be too, simliar
144 -- for smaller than l - d.
145 -- Add a little extra to keep clear
146 -- from the boundary cases.
150 | x == 0 && y > 0 = pi/2
151 | x < 0 && y > 0 = pi + atan (y/x)
152 |(x <= 0 && y < 0) ||
153 (x < 0 && isNegativeZero y) ||
154 (isNegativeZero x && isNegativeZero y)
156 | y == 0 && (x < 0 || isNegativeZero x)
157 = pi -- must be after the previous test on zero y
158 | x==0 && y==0 = y -- must be after the other double zero tests
159 | otherwise = x + y -- x or y is a NaN, return a NaN (via +)
163 %*********************************************************
165 \subsection{Type @Float@}
167 %*********************************************************
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 = 0.5 * log ((1.0+x) / (1.0-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 + clamp bf k)
283 where bf = FLT_MAX_EXP - (FLT_MIN_EXP) + 4*FLT_MANT_DIG
285 isNaN x = 0 /= isFloatNaN x
286 isInfinite x = 0 /= isFloatInfinite x
287 isDenormalized x = 0 /= isFloatDenormalized x
288 isNegativeZero x = 0 /= isFloatNegativeZero x
291 instance Show Float where
292 showsPrec x = showSignedFloat showFloat x
293 showList = showList__ (showsPrec 0)
296 %*********************************************************
298 \subsection{Type @Double@}
300 %*********************************************************
303 instance Num Double where
304 (+) x y = plusDouble x y
305 (-) x y = minusDouble x y
306 negate x = negateDouble x
307 (*) x y = timesDouble x y
309 | otherwise = negateDouble x
310 signum x | x == 0.0 = 0
312 | otherwise = negate 1
314 {-# INLINE fromInteger #-}
315 fromInteger i = D# (doubleFromInteger i)
318 instance Real Double where
319 toRational x = (m%1)*(b%1)^^n
320 where (m,n) = decodeFloat x
323 instance Fractional Double where
324 (/) x y = divideDouble x y
325 fromRational x = fromRat x
328 instance Floating Double where
329 pi = 3.141592653589793238
332 sqrt x = sqrtDouble x
336 asin x = asinDouble x
337 acos x = acosDouble x
338 atan x = atanDouble x
339 sinh x = sinhDouble x
340 cosh x = coshDouble x
341 tanh x = tanhDouble x
342 (**) x y = powerDouble x y
343 logBase x y = log y / log x
345 asinh x = log (x + sqrt (1.0+x*x))
346 acosh x = log (x + (x+1.0) * sqrt ((x-1.0)/(x+1.0)))
347 atanh x = 0.5 * log ((1.0+x) / (1.0-x))
349 {-# RULES "truncate/Double->Int" truncate = double2Int #-}
350 instance RealFrac Double where
352 {-# SPECIALIZE properFraction :: Double -> (Int, Double) #-}
353 {-# SPECIALIZE round :: Double -> Int #-}
355 {-# SPECIALIZE properFraction :: Double -> (Integer, Double) #-}
356 {-# SPECIALIZE round :: Double -> Integer #-}
358 -- ceiling, floor, and truncate are all small
359 {-# INLINE ceiling #-}
361 {-# INLINE truncate #-}
364 = case (decodeFloat x) of { (m,n) ->
365 let b = floatRadix x in
367 (fromInteger m * fromInteger b ^ n, 0.0)
369 case (quotRem m (b^(negate n))) of { (w,r) ->
370 (fromInteger w, encodeFloat r n)
374 truncate x = case properFraction x of
377 round x = case properFraction x of
379 m = if r < 0.0 then n - 1 else n + 1
380 half_down = abs r - 0.5
382 case (compare half_down 0.0) of
384 EQ -> if even n then n else m
387 ceiling x = case properFraction x of
388 (n,r) -> if r > 0.0 then n + 1 else n
390 floor x = case properFraction x of
391 (n,r) -> if r < 0.0 then n - 1 else n
393 instance RealFloat Double where
394 floatRadix _ = FLT_RADIX -- from float.h
395 floatDigits _ = DBL_MANT_DIG -- ditto
396 floatRange _ = (DBL_MIN_EXP, DBL_MAX_EXP) -- ditto
399 = case decodeDoubleInteger x# of
400 (# i, j #) -> (i, I# j)
402 encodeFloat i (I# j) = D# (encodeDoubleInteger i j)
404 exponent x = case decodeFloat x of
405 (m,n) -> if m == 0 then 0 else n + floatDigits x
407 significand x = case decodeFloat x of
408 (m,_) -> encodeFloat m (negate (floatDigits x))
410 scaleFloat k x = case decodeFloat x of
411 (m,n) -> encodeFloat m (n + clamp bd k)
412 where bd = DBL_MAX_EXP - (DBL_MIN_EXP) + 4*DBL_MANT_DIG
414 isNaN x = 0 /= isDoubleNaN x
415 isInfinite x = 0 /= isDoubleInfinite x
416 isDenormalized x = 0 /= isDoubleDenormalized x
417 isNegativeZero x = 0 /= isDoubleNegativeZero x
420 instance Show Double where
421 showsPrec x = showSignedFloat showFloat x
422 showList = showList__ (showsPrec 0)
425 %*********************************************************
427 \subsection{@Enum@ instances}
429 %*********************************************************
431 The @Enum@ instances for Floats and Doubles are slightly unusual.
432 The @toEnum@ function truncates numbers to Int. The definitions
433 of @enumFrom@ and @enumFromThen@ allow floats to be used in arithmetic
434 series: [0,0.1 .. 1.0]. However, roundoff errors make these somewhat
435 dubious. This example may have either 10 or 11 elements, depending on
436 how 0.1 is represented.
438 NOTE: The instances for Float and Double do not make use of the default
439 methods for @enumFromTo@ and @enumFromThenTo@, as these rely on there being
440 a `non-lossy' conversion to and from Ints. Instead we make use of the
441 1.2 default methods (back in the days when Enum had Ord as a superclass)
442 for these (@numericEnumFromTo@ and @numericEnumFromThenTo@ below.)
445 instance Enum Float where
449 fromEnum = fromInteger . truncate -- may overflow
450 enumFrom = numericEnumFrom
451 enumFromTo = numericEnumFromTo
452 enumFromThen = numericEnumFromThen
453 enumFromThenTo = numericEnumFromThenTo
455 instance Enum Double where
459 fromEnum = fromInteger . truncate -- may overflow
460 enumFrom = numericEnumFrom
461 enumFromTo = numericEnumFromTo
462 enumFromThen = numericEnumFromThen
463 enumFromThenTo = numericEnumFromThenTo
467 %*********************************************************
469 \subsection{Printing floating point}
471 %*********************************************************
475 -- | Show a signed 'RealFloat' value to full precision
476 -- using standard decimal notation for arguments whose absolute value lies
477 -- between @0.1@ and @9,999,999@, and scientific notation otherwise.
478 showFloat :: (RealFloat a) => a -> ShowS
479 showFloat x = showString (formatRealFloat FFGeneric Nothing x)
481 -- These are the format types. This type is not exported.
483 data FFFormat = FFExponent | FFFixed | FFGeneric
485 formatRealFloat :: (RealFloat a) => FFFormat -> Maybe Int -> a -> String
486 formatRealFloat fmt decs x
488 | isInfinite x = if x < 0 then "-Infinity" else "Infinity"
489 | x < 0 || isNegativeZero x = '-':doFmt fmt (floatToDigits (toInteger base) (-x))
490 | otherwise = doFmt fmt (floatToDigits (toInteger base) x)
494 doFmt format (is, e) =
495 let ds = map intToDigit is in
498 doFmt (if e < 0 || e > 7 then FFExponent else FFFixed)
503 let show_e' = show (e-1) in
506 [d] -> d : ".0e" ++ show_e'
507 (d:ds') -> d : '.' : ds' ++ "e" ++ show_e'
508 [] -> error "formatRealFloat/doFmt/FFExponent: []"
510 let dec' = max dec 1 in
512 [0] -> '0' :'.' : take dec' (repeat '0') ++ "e0"
515 (ei,is') = roundTo base (dec'+1) is
516 (d:ds') = map intToDigit (if ei > 0 then init is' else is')
518 d:'.':ds' ++ 'e':show (e-1+ei)
521 mk0 ls = case ls of { "" -> "0" ; _ -> ls}
525 | e <= 0 -> "0." ++ replicate (-e) '0' ++ ds
528 f 0 s rs = mk0 (reverse s) ++ '.':mk0 rs
529 f n s "" = f (n-1) ('0':s) ""
530 f n s (r:rs) = f (n-1) (r:s) rs
534 let dec' = max dec 0 in
537 (ei,is') = roundTo base (dec' + e) is
538 (ls,rs) = splitAt (e+ei) (map intToDigit is')
540 mk0 ls ++ (if null rs then "" else '.':rs)
543 (ei,is') = roundTo base dec' (replicate (-e) 0 ++ is)
544 d:ds' = map intToDigit (if ei > 0 then is' else 0:is')
546 d : (if null ds' then "" else '.':ds')
549 roundTo :: Int -> Int -> [Int] -> (Int,[Int])
554 _ -> error "roundTo: bad Value"
558 f n [] = (0, replicate n 0)
559 f 0 (x:_) = (if x >= b2 then 1 else 0, [])
561 | i' == base = (1,0:ds)
562 | otherwise = (0,i':ds)
567 -- Based on "Printing Floating-Point Numbers Quickly and Accurately"
568 -- by R.G. Burger and R.K. Dybvig in PLDI 96.
569 -- This version uses a much slower logarithm estimator. It should be improved.
571 -- | 'floatToDigits' takes a base and a non-negative 'RealFloat' number,
572 -- and returns a list of digits and an exponent.
573 -- In particular, if @x>=0@, and
575 -- > floatToDigits base x = ([d1,d2,...,dn], e)
581 -- (2) @x = 0.d1d2...dn * (base**e)@
583 -- (3) @0 <= di <= base-1@
585 floatToDigits :: (RealFloat a) => Integer -> a -> ([Int], Int)
586 floatToDigits _ 0 = ([0], 0)
587 floatToDigits base x =
589 (f0, e0) = decodeFloat x
590 (minExp0, _) = floatRange x
593 minExp = minExp0 - p -- the real minimum exponent
594 -- Haskell requires that f be adjusted so denormalized numbers
595 -- will have an impossibly low exponent. Adjust for this.
597 let n = minExp - e0 in
598 if n > 0 then (f0 `quot` (expt b n), e0+n) else (f0, e0)
602 if f == expt b (p-1) then
603 (f*be*b*2, 2*b, be*b, be) -- according to Burger and Dybvig
607 if e > minExp && f == expt b (p-1) then
608 (f*b*2, expt b (-e+1)*2, b, 1)
610 (f*2, expt b (-e)*2, 1, 1)
616 if b == 2 && base == 10 then
617 -- logBase 10 2 is very slightly larger than 8651/28738
618 -- (about 5.3558e-10), so if log x >= 0, the approximation
619 -- k1 is too small, hence we add one and need one fixup step less.
620 -- If log x < 0, the approximation errs rather on the high side.
621 -- That is usually more than compensated for by ignoring the
622 -- fractional part of logBase 2 x, but when x is a power of 1/2
623 -- or slightly larger and the exponent is a multiple of the
624 -- denominator of the rational approximation to logBase 10 2,
625 -- k1 is larger than logBase 10 x. If k1 > 1 + logBase 10 x,
626 -- we get a leading zero-digit we don't want.
627 -- With the approximation 3/10, this happened for
628 -- 0.5^1030, 0.5^1040, ..., 0.5^1070 and values close above.
629 -- The approximation 8651/28738 guarantees k1 < 1 + logBase 10 x
630 -- for IEEE-ish floating point types with exponent fields
631 -- <= 17 bits and mantissae of several thousand bits, earlier
632 -- convergents to logBase 10 2 would fail for long double.
633 -- Using quot instead of div is a little faster and requires
634 -- fewer fixup steps for negative lx.
636 k1 = (lx * 8651) `quot` 28738
637 in if lx >= 0 then k1 + 1 else k1
639 -- f :: Integer, log :: Float -> Float,
640 -- ceiling :: Float -> Int
641 ceiling ((log (fromInteger (f+1) :: Float) +
642 fromIntegral 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) `quotRem` 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 = -1/0 -- -Infinity
742 | otherwise = 0/0 -- NaN
744 fromRat (n :% d) | n > 0 = fromRat' (n :% d)
745 | n < 0 = - fromRat' ((-n) :% d)
746 | otherwise = encodeFloat 0 0 -- Zero
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
788 if base == 10 && n <= maxExpt10 then
793 expts :: Array Int Integer
794 expts = array (minExpt,maxExpt) [(n,2^n) | n <- [minExpt .. maxExpt]]
799 expts10 :: Array Int Integer
800 expts10 = array (minExpt,maxExpt10) [(n,10^n) | n <- [minExpt .. maxExpt10]]
802 -- Compute the (floor of the) log of i in base b.
803 -- Simplest way would be just divide i by b until it's smaller then b, but that would
804 -- be very slow! We are just slightly more clever.
805 integerLogBase :: Integer -> Integer -> Int
808 | otherwise = doDiv (i `div` (b^l)) l
810 -- Try squaring the base first to cut down the number of divisions.
811 l = 2 * integerLogBase (b*b) i
813 doDiv :: Integer -> Int -> Int
816 | otherwise = doDiv (x `div` b) (y+1)
821 %*********************************************************
823 \subsection{Floating point numeric primops}
825 %*********************************************************
827 Definitions of the boxed PrimOps; these will be
828 used in the case of partial applications, etc.
831 plusFloat, minusFloat, timesFloat, divideFloat :: Float -> Float -> Float
832 plusFloat (F# x) (F# y) = F# (plusFloat# x y)
833 minusFloat (F# x) (F# y) = F# (minusFloat# x y)
834 timesFloat (F# x) (F# y) = F# (timesFloat# x y)
835 divideFloat (F# x) (F# y) = F# (divideFloat# x y)
837 negateFloat :: Float -> Float
838 negateFloat (F# x) = F# (negateFloat# x)
840 gtFloat, geFloat, eqFloat, neFloat, ltFloat, leFloat :: Float -> Float -> Bool
841 gtFloat (F# x) (F# y) = gtFloat# x y
842 geFloat (F# x) (F# y) = geFloat# x y
843 eqFloat (F# x) (F# y) = eqFloat# x y
844 neFloat (F# x) (F# y) = neFloat# x y
845 ltFloat (F# x) (F# y) = ltFloat# x y
846 leFloat (F# x) (F# y) = leFloat# x y
848 float2Int :: Float -> Int
849 float2Int (F# x) = I# (float2Int# x)
851 int2Float :: Int -> Float
852 int2Float (I# x) = F# (int2Float# x)
854 expFloat, logFloat, sqrtFloat :: Float -> Float
855 sinFloat, cosFloat, tanFloat :: Float -> Float
856 asinFloat, acosFloat, atanFloat :: Float -> Float
857 sinhFloat, coshFloat, tanhFloat :: Float -> Float
858 expFloat (F# x) = F# (expFloat# x)
859 logFloat (F# x) = F# (logFloat# x)
860 sqrtFloat (F# x) = F# (sqrtFloat# x)
861 sinFloat (F# x) = F# (sinFloat# x)
862 cosFloat (F# x) = F# (cosFloat# x)
863 tanFloat (F# x) = F# (tanFloat# x)
864 asinFloat (F# x) = F# (asinFloat# x)
865 acosFloat (F# x) = F# (acosFloat# x)
866 atanFloat (F# x) = F# (atanFloat# x)
867 sinhFloat (F# x) = F# (sinhFloat# x)
868 coshFloat (F# x) = F# (coshFloat# x)
869 tanhFloat (F# x) = F# (tanhFloat# x)
871 powerFloat :: Float -> Float -> Float
872 powerFloat (F# x) (F# y) = F# (powerFloat# x y)
874 -- definitions of the boxed PrimOps; these will be
875 -- used in the case of partial applications, etc.
877 plusDouble, minusDouble, timesDouble, divideDouble :: Double -> Double -> Double
878 plusDouble (D# x) (D# y) = D# (x +## y)
879 minusDouble (D# x) (D# y) = D# (x -## y)
880 timesDouble (D# x) (D# y) = D# (x *## y)
881 divideDouble (D# x) (D# y) = D# (x /## y)
883 negateDouble :: Double -> Double
884 negateDouble (D# x) = D# (negateDouble# x)
886 gtDouble, geDouble, eqDouble, neDouble, leDouble, ltDouble :: Double -> Double -> Bool
887 gtDouble (D# x) (D# y) = x >## y
888 geDouble (D# x) (D# y) = x >=## y
889 eqDouble (D# x) (D# y) = x ==## y
890 neDouble (D# x) (D# y) = x /=## y
891 ltDouble (D# x) (D# y) = x <## y
892 leDouble (D# x) (D# y) = x <=## y
894 double2Int :: Double -> Int
895 double2Int (D# x) = I# (double2Int# x)
897 int2Double :: Int -> Double
898 int2Double (I# x) = D# (int2Double# x)
900 double2Float :: Double -> Float
901 double2Float (D# x) = F# (double2Float# x)
903 float2Double :: Float -> Double
904 float2Double (F# x) = D# (float2Double# x)
906 expDouble, logDouble, sqrtDouble :: Double -> Double
907 sinDouble, cosDouble, tanDouble :: Double -> Double
908 asinDouble, acosDouble, atanDouble :: Double -> Double
909 sinhDouble, coshDouble, tanhDouble :: Double -> Double
910 expDouble (D# x) = D# (expDouble# x)
911 logDouble (D# x) = D# (logDouble# x)
912 sqrtDouble (D# x) = D# (sqrtDouble# x)
913 sinDouble (D# x) = D# (sinDouble# x)
914 cosDouble (D# x) = D# (cosDouble# x)
915 tanDouble (D# x) = D# (tanDouble# x)
916 asinDouble (D# x) = D# (asinDouble# x)
917 acosDouble (D# x) = D# (acosDouble# x)
918 atanDouble (D# x) = D# (atanDouble# x)
919 sinhDouble (D# x) = D# (sinhDouble# x)
920 coshDouble (D# x) = D# (coshDouble# x)
921 tanhDouble (D# x) = D# (tanhDouble# x)
923 powerDouble :: Double -> Double -> Double
924 powerDouble (D# x) (D# y) = D# (x **## y)
928 foreign import ccall unsafe "isFloatNaN" isFloatNaN :: Float -> Int
929 foreign import ccall unsafe "isFloatInfinite" isFloatInfinite :: Float -> Int
930 foreign import ccall unsafe "isFloatDenormalized" isFloatDenormalized :: Float -> Int
931 foreign import ccall unsafe "isFloatNegativeZero" isFloatNegativeZero :: Float -> Int
934 foreign import ccall unsafe "isDoubleNaN" isDoubleNaN :: Double -> Int
935 foreign import ccall unsafe "isDoubleInfinite" isDoubleInfinite :: Double -> Int
936 foreign import ccall unsafe "isDoubleDenormalized" isDoubleDenormalized :: Double -> Int
937 foreign import ccall unsafe "isDoubleNegativeZero" isDoubleNegativeZero :: Double -> Int
940 %*********************************************************
942 \subsection{Coercion rules}
944 %*********************************************************
948 "fromIntegral/Int->Float" fromIntegral = int2Float
949 "fromIntegral/Int->Double" fromIntegral = int2Double
950 "realToFrac/Float->Float" realToFrac = id :: Float -> Float
951 "realToFrac/Float->Double" realToFrac = float2Double
952 "realToFrac/Double->Float" realToFrac = double2Float
953 "realToFrac/Double->Double" realToFrac = id :: Double -> Double
954 "realToFrac/Int->Double" realToFrac = int2Double -- See Note [realToFrac int-to-float]
955 "realToFrac/Int->Float" realToFrac = int2Float -- ..ditto
959 Note [realToFrac int-to-float]
960 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
961 Don found that the RULES for realToFrac/Int->Double and simliarly
962 Float made a huge difference to some stream-fusion programs. Here's
965 import Data.Array.Vector
970 let c = replicateU n (2::Double)
971 a = mapU realToFrac (enumFromToU 0 (n-1) ) :: UArr Double
972 print (sumU (zipWithU (*) c a))
974 Without the RULE we get this loop body:
976 case $wtoRational sc_sY4 of ww_aM7 { (# ww1_aM9, ww2_aMa #) ->
977 case $wfromRat ww1_aM9 ww2_aMa of tpl_X1P { D# ipv_sW3 ->
981 (+## sc2_sY6 (*## 2.0 ipv_sW3))
988 (+## sc2_sXV (*## 2.0 (int2Double# sc_sXT)))
990 The running time of the program goes from 120 seconds to 0.198 seconds
991 with the native backend, and 0.143 seconds with the C backend.
993 A few more details in Trac #2251, and the patch message
994 "Add RULES for realToFrac from Int".
996 %*********************************************************
1000 %*********************************************************
1003 showSignedFloat :: (RealFloat a)
1004 => (a -> ShowS) -- ^ a function that can show unsigned values
1005 -> Int -- ^ the precedence of the enclosing context
1006 -> a -- ^ the value to show
1008 showSignedFloat showPos p x
1009 | x < 0 || isNegativeZero x
1010 = showParen (p > 6) (showChar '-' . showPos (-x))
1011 | otherwise = showPos x
1014 We need to prevent over/underflow of the exponent in encodeFloat when
1015 called from scaleFloat, hence we clamp the scaling parameter.
1016 We must have a large enough range to cover the maximum difference of
1017 exponents returned by decodeFloat.
1019 clamp :: Int -> Int -> Int
1020 clamp bd k = max (-bd) (min bd k)