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 (S# i#) = case (int2Float# i#) of { d# -> F# d# }
193 fromInteger (J# s# d#) = encodeFloat# s# d# 0
194 -- previous code: fromInteger n = encodeFloat n 0
195 -- doesn't work too well, because encodeFloat is defined in
196 -- terms of ccalls which can never be simplified away. We
197 -- want simple literals like (fromInteger 3 :: Float) to turn
198 -- into (F# 3.0), hence the special case for S# here.
200 instance Real Float where
201 toRational x = (m%1)*(b%1)^^n
202 where (m,n) = decodeFloat x
205 instance Fractional Float where
206 (/) x y = divideFloat x y
207 fromRational x = fromRat x
210 {-# RULES "truncate/Float->Int" truncate = float2Int #-}
211 instance RealFrac Float where
213 {-# SPECIALIZE properFraction :: Float -> (Int, Float) #-}
214 {-# SPECIALIZE round :: Float -> Int #-}
216 {-# SPECIALIZE properFraction :: Float -> (Integer, Float) #-}
217 {-# SPECIALIZE round :: Float -> Integer #-}
219 -- ceiling, floor, and truncate are all small
220 {-# INLINE ceiling #-}
222 {-# INLINE truncate #-}
225 = case (decodeFloat x) of { (m,n) ->
226 let b = floatRadix x in
228 (fromInteger m * fromInteger b ^ n, 0.0)
230 case (quotRem m (b^(negate n))) of { (w,r) ->
231 (fromInteger w, encodeFloat r n)
235 truncate x = case properFraction x of
238 round x = case properFraction x of
240 m = if r < 0.0 then n - 1 else n + 1
241 half_down = abs r - 0.5
243 case (compare half_down 0.0) of
245 EQ -> if even n then n else m
248 ceiling x = case properFraction x of
249 (n,r) -> if r > 0.0 then n + 1 else n
251 floor x = case properFraction x of
252 (n,r) -> if r < 0.0 then n - 1 else n
254 instance Floating Float where
255 pi = 3.141592653589793238
268 (**) x y = powerFloat x y
269 logBase x y = log y / log x
271 asinh x = log (x + sqrt (1.0+x*x))
272 acosh x = log (x + (x+1.0) * sqrt ((x-1.0)/(x+1.0)))
273 atanh x = log ((x+1.0) / sqrt (1.0-x*x))
275 instance RealFloat Float where
276 floatRadix _ = FLT_RADIX -- from float.h
277 floatDigits _ = FLT_MANT_DIG -- ditto
278 floatRange _ = (FLT_MIN_EXP, FLT_MAX_EXP) -- ditto
281 = case decodeFloat# f# of
282 (# exp#, s#, d# #) -> (J# s# d#, I# exp#)
284 encodeFloat (S# i) j = int_encodeFloat# i j
285 encodeFloat (J# s# d#) e = encodeFloat# s# d# e
287 exponent x = case decodeFloat x of
288 (m,n) -> if m == 0 then 0 else n + floatDigits x
290 significand x = case decodeFloat x of
291 (m,_) -> encodeFloat m (negate (floatDigits x))
293 scaleFloat k x = case decodeFloat x of
294 (m,n) -> encodeFloat m (n+k)
295 isNaN x = 0 /= isFloatNaN x
296 isInfinite x = 0 /= isFloatInfinite x
297 isDenormalized x = 0 /= isFloatDenormalized x
298 isNegativeZero x = 0 /= isFloatNegativeZero x
301 instance Show Float where
302 showsPrec x = showSigned showFloat x
303 showList = showList__ (showsPrec 0)
306 %*********************************************************
308 \subsection{Type @Double@}
310 %*********************************************************
313 instance Eq Double where
314 (D# x) == (D# y) = x ==## y
316 instance Ord Double where
317 (D# x) `compare` (D# y) | x <## y = LT
321 (D# x) < (D# y) = x <## y
322 (D# x) <= (D# y) = x <=## y
323 (D# x) >= (D# y) = x >=## y
324 (D# x) > (D# y) = x >## y
326 instance Num Double where
327 (+) x y = plusDouble x y
328 (-) x y = minusDouble x y
329 negate x = negateDouble x
330 (*) x y = timesDouble x y
332 | otherwise = negateDouble x
333 signum x | x == 0.0 = 0
335 | otherwise = negate 1
337 {-# INLINE fromInteger #-}
338 -- See comments with Num Float
339 fromInteger (S# i#) = case (int2Double# i#) of { d# -> D# d# }
340 fromInteger (J# s# d#) = encodeDouble# s# d# 0
343 instance Real Double where
344 toRational x = (m%1)*(b%1)^^n
345 where (m,n) = decodeFloat x
348 instance Fractional Double where
349 (/) x y = divideDouble x y
350 fromRational x = fromRat x
353 instance Floating Double where
354 pi = 3.141592653589793238
357 sqrt x = sqrtDouble x
361 asin x = asinDouble x
362 acos x = acosDouble x
363 atan x = atanDouble x
364 sinh x = sinhDouble x
365 cosh x = coshDouble x
366 tanh x = tanhDouble x
367 (**) x y = powerDouble x y
368 logBase x y = log y / log x
370 asinh x = log (x + sqrt (1.0+x*x))
371 acosh x = log (x + (x+1.0) * sqrt ((x-1.0)/(x+1.0)))
372 atanh x = log ((x+1.0) / sqrt (1.0-x*x))
374 {-# RULES "truncate/Double->Int" truncate = double2Int #-}
375 instance RealFrac Double where
377 {-# SPECIALIZE properFraction :: Double -> (Int, Double) #-}
378 {-# SPECIALIZE round :: Double -> Int #-}
380 {-# SPECIALIZE properFraction :: Double -> (Integer, Double) #-}
381 {-# SPECIALIZE round :: Double -> Integer #-}
383 -- ceiling, floor, and truncate are all small
384 {-# INLINE ceiling #-}
386 {-# INLINE truncate #-}
389 = case (decodeFloat x) of { (m,n) ->
390 let b = floatRadix x in
392 (fromInteger m * fromInteger b ^ n, 0.0)
394 case (quotRem m (b^(negate n))) of { (w,r) ->
395 (fromInteger w, encodeFloat r n)
399 truncate x = case properFraction x of
402 round x = case properFraction x of
404 m = if r < 0.0 then n - 1 else n + 1
405 half_down = abs r - 0.5
407 case (compare half_down 0.0) of
409 EQ -> if even n then n else m
412 ceiling x = case properFraction x of
413 (n,r) -> if r > 0.0 then n + 1 else n
415 floor x = case properFraction x of
416 (n,r) -> if r < 0.0 then n - 1 else n
418 instance RealFloat Double where
419 floatRadix _ = FLT_RADIX -- from float.h
420 floatDigits _ = DBL_MANT_DIG -- ditto
421 floatRange _ = (DBL_MIN_EXP, DBL_MAX_EXP) -- ditto
424 = case decodeDouble# x# of
425 (# exp#, s#, d# #) -> (J# s# d#, I# exp#)
427 encodeFloat (S# i) j = int_encodeDouble# i j
428 encodeFloat (J# s# d#) e = encodeDouble# s# d# e
430 exponent x = case decodeFloat x of
431 (m,n) -> if m == 0 then 0 else n + floatDigits x
433 significand x = case decodeFloat x of
434 (m,_) -> encodeFloat m (negate (floatDigits x))
436 scaleFloat k x = case decodeFloat x of
437 (m,n) -> encodeFloat m (n+k)
439 isNaN x = 0 /= isDoubleNaN x
440 isInfinite x = 0 /= isDoubleInfinite x
441 isDenormalized x = 0 /= isDoubleDenormalized x
442 isNegativeZero x = 0 /= isDoubleNegativeZero x
445 instance Show Double where
446 showsPrec x = showSigned showFloat x
447 showList = showList__ (showsPrec 0)
450 %*********************************************************
452 \subsection{@Enum@ instances}
454 %*********************************************************
456 The @Enum@ instances for Floats and Doubles are slightly unusual.
457 The @toEnum@ function truncates numbers to Int. The definitions
458 of @enumFrom@ and @enumFromThen@ allow floats to be used in arithmetic
459 series: [0,0.1 .. 1.0]. However, roundoff errors make these somewhat
460 dubious. This example may have either 10 or 11 elements, depending on
461 how 0.1 is represented.
463 NOTE: The instances for Float and Double do not make use of the default
464 methods for @enumFromTo@ and @enumFromThenTo@, as these rely on there being
465 a `non-lossy' conversion to and from Ints. Instead we make use of the
466 1.2 default methods (back in the days when Enum had Ord as a superclass)
467 for these (@numericEnumFromTo@ and @numericEnumFromThenTo@ below.)
470 instance Enum Float where
474 fromEnum = fromInteger . truncate -- may overflow
475 enumFrom = numericEnumFrom
476 enumFromTo = numericEnumFromTo
477 enumFromThen = numericEnumFromThen
478 enumFromThenTo = numericEnumFromThenTo
480 instance Enum Double where
484 fromEnum = fromInteger . truncate -- may overflow
485 enumFrom = numericEnumFrom
486 enumFromTo = numericEnumFromTo
487 enumFromThen = numericEnumFromThen
488 enumFromThenTo = numericEnumFromThenTo
492 %*********************************************************
494 \subsection{Printing floating point}
496 %*********************************************************
500 -- | Show a signed 'RealFloat' value to full precision
501 -- using standard decimal notation for arguments whose absolute value lies
502 -- between @0.1@ and @9,999,999@, and scientific notation otherwise.
503 showFloat :: (RealFloat a) => a -> ShowS
504 showFloat x = showString (formatRealFloat FFGeneric Nothing x)
506 -- These are the format types. This type is not exported.
508 data FFFormat = FFExponent | FFFixed | FFGeneric
510 formatRealFloat :: (RealFloat a) => FFFormat -> Maybe Int -> a -> String
511 formatRealFloat fmt decs x
513 | isInfinite x = if x < 0 then "-Infinity" else "Infinity"
514 | x < 0 || isNegativeZero x = '-':doFmt fmt (floatToDigits (toInteger base) (-x))
515 | otherwise = doFmt fmt (floatToDigits (toInteger base) x)
519 doFmt format (is, e) =
520 let ds = map intToDigit is in
523 doFmt (if e < 0 || e > 7 then FFExponent else FFFixed)
528 let show_e' = show (e-1) in
531 [d] -> d : ".0e" ++ show_e'
532 (d:ds') -> d : '.' : ds' ++ "e" ++ show_e'
534 let dec' = max dec 1 in
536 [0] -> '0' :'.' : take dec' (repeat '0') ++ "e0"
539 (ei,is') = roundTo base (dec'+1) is
540 (d:ds') = map intToDigit (if ei > 0 then init is' else is')
542 d:'.':ds' ++ 'e':show (e-1+ei)
545 mk0 ls = case ls of { "" -> "0" ; _ -> ls}
549 | e <= 0 -> "0." ++ replicate (-e) '0' ++ ds
552 f 0 s rs = mk0 (reverse s) ++ '.':mk0 rs
553 f n s "" = f (n-1) ('0':s) ""
554 f n s (r:rs) = f (n-1) (r:s) rs
558 let dec' = max dec 0 in
561 (ei,is') = roundTo base (dec' + e) is
562 (ls,rs) = splitAt (e+ei) (map intToDigit is')
564 mk0 ls ++ (if null rs then "" else '.':rs)
567 (ei,is') = roundTo base dec' (replicate (-e) 0 ++ is)
568 d:ds' = map intToDigit (if ei > 0 then is' else 0:is')
570 d : (if null ds' then "" else '.':ds')
573 roundTo :: Int -> Int -> [Int] -> (Int,[Int])
581 f n [] = (0, replicate n 0)
582 f 0 (x:_) = (if x >= b2 then 1 else 0, [])
584 | i' == base = (1,0:ds)
585 | otherwise = (0,i':ds)
590 -- Based on "Printing Floating-Point Numbers Quickly and Accurately"
591 -- by R.G. Burger and R.K. Dybvig in PLDI 96.
592 -- This version uses a much slower logarithm estimator. It should be improved.
594 -- | 'floatToDigits' takes a base and a non-negative 'RealFloat' number,
595 -- and returns a list of digits and an exponent.
596 -- In particular, if @x>=0@, and
598 -- > floatToDigits base x = ([d1,d2,...,dn], e)
604 -- (2) @x = 0.d1d2...dn * (base**e)@
606 -- (3) @0 <= di <= base-1@
608 floatToDigits :: (RealFloat a) => Integer -> a -> ([Int], Int)
609 floatToDigits _ 0 = ([0], 0)
610 floatToDigits base x =
612 (f0, e0) = decodeFloat x
613 (minExp0, _) = floatRange x
616 minExp = minExp0 - p -- the real minimum exponent
617 -- Haskell requires that f be adjusted so denormalized numbers
618 -- will have an impossibly low exponent. Adjust for this.
620 let n = minExp - e0 in
621 if n > 0 then (f0 `div` (b^n), e0+n) else (f0, e0)
626 (f*be*b*2, 2*b, be*b, b)
630 if e > minExp && f == b^(p-1) then
631 (f*b*2, b^(-e+1)*2, b, 1)
633 (f*2, b^(-e)*2, 1, 1)
637 if b == 2 && base == 10 then
638 -- logBase 10 2 is slightly bigger than 3/10 so
639 -- the following will err on the low side. Ignoring
640 -- the fraction will make it err even more.
641 -- Haskell promises that p-1 <= logBase b f < p.
642 (p - 1 + e0) * 3 `div` 10
644 ceiling ((log (fromInteger (f+1)) +
645 fromInteger (int2Integer e) * log (fromInteger b)) /
646 log (fromInteger base))
647 --WAS: fromInt e * log (fromInteger b))
651 if r + mUp <= expt base n * s then n else fixup (n+1)
653 if expt base (-n) * (r + mUp) <= s then n else fixup (n+1)
657 gen ds rn sN mUpN mDnN =
659 (dn, rn') = (rn * base) `divMod` sN
663 case (rn' < mDnN', rn' + mUpN' > sN) of
664 (True, False) -> dn : ds
665 (False, True) -> dn+1 : ds
666 (True, True) -> if rn' * 2 < sN then dn : ds else dn+1 : ds
667 (False, False) -> gen (dn:ds) rn' sN mUpN' mDnN'
671 gen [] r (s * expt base k) mUp mDn
673 let bk = expt base (-k) in
674 gen [] (r * bk) s (mUp * bk) (mDn * bk)
676 (map fromIntegral (reverse rds), k)
681 %*********************************************************
683 \subsection{Converting from a Rational to a RealFloat
685 %*********************************************************
687 [In response to a request for documentation of how fromRational works,
688 Joe Fasel writes:] A quite reasonable request! This code was added to
689 the Prelude just before the 1.2 release, when Lennart, working with an
690 early version of hbi, noticed that (read . show) was not the identity
691 for floating-point numbers. (There was a one-bit error about half the
692 time.) The original version of the conversion function was in fact
693 simply a floating-point divide, as you suggest above. The new version
694 is, I grant you, somewhat denser.
696 Unfortunately, Joe's code doesn't work! Here's an example:
698 main = putStr (shows (1.82173691287639817263897126389712638972163e-300::Double) "\n")
703 1.8217369128763981e-300
708 fromRat :: (RealFloat a) => Rational -> a
712 -- If the exponent of the nearest floating-point number to x
713 -- is e, then the significand is the integer nearest xb^(-e),
714 -- where b is the floating-point radix. We start with a good
715 -- guess for e, and if it is correct, the exponent of the
716 -- floating-point number we construct will again be e. If
717 -- not, one more iteration is needed.
719 f e = if e' == e then y else f e'
720 where y = encodeFloat (round (x * (1 % b)^^e)) e
721 (_,e') = decodeFloat y
724 -- We obtain a trial exponent by doing a floating-point
725 -- division of x's numerator by its denominator. The
726 -- result of this division may not itself be the ultimate
727 -- result, because of an accumulation of three rounding
730 (s,e) = decodeFloat (fromInteger (numerator x) `asTypeOf` x'
731 / fromInteger (denominator x))
734 Now, here's Lennart's code (which works)
737 -- | Converts a 'Rational' value into any type in class 'RealFloat'.
738 {-# SPECIALISE fromRat :: Rational -> Double,
739 Rational -> Float #-}
740 fromRat :: (RealFloat a) => Rational -> a
742 -- Deal with special cases first, delegating the real work to fromRat'
743 fromRat (n :% 0) | n > 0 = 1/0 -- +Infinity
744 | n == 0 = 0/0 -- NaN
745 | n < 0 = -1/0 -- -Infinity
747 fromRat (n :% d) | n > 0 = fromRat' (n :% d)
748 | n == 0 = encodeFloat 0 0 -- Zero
749 | n < 0 = - fromRat' ((-n) :% d)
751 -- Conversion process:
752 -- Scale the rational number by the RealFloat base until
753 -- it lies in the range of the mantissa (as used by decodeFloat/encodeFloat).
754 -- Then round the rational to an Integer and encode it with the exponent
755 -- that we got from the scaling.
756 -- To speed up the scaling process we compute the log2 of the number to get
757 -- a first guess of the exponent.
759 fromRat' :: (RealFloat a) => Rational -> a
760 -- Invariant: argument is strictly positive
762 where b = floatRadix r
764 (minExp0, _) = floatRange r
765 minExp = minExp0 - p -- the real minimum exponent
766 xMin = toRational (expt b (p-1))
767 xMax = toRational (expt b p)
768 p0 = (integerLogBase b (numerator x) - integerLogBase b (denominator x) - p) `max` minExp
769 f = if p0 < 0 then 1 % expt b (-p0) else expt b p0 % 1
770 (x', p') = scaleRat (toRational b) minExp xMin xMax p0 (x / f)
771 r = encodeFloat (round x') p'
773 -- Scale x until xMin <= x < xMax, or p (the exponent) <= minExp.
774 scaleRat :: Rational -> Int -> Rational -> Rational -> Int -> Rational -> (Rational, Int)
775 scaleRat b minExp xMin xMax p x
776 | p <= minExp = (x, p)
777 | x >= xMax = scaleRat b minExp xMin xMax (p+1) (x/b)
778 | x < xMin = scaleRat b minExp xMin xMax (p-1) (x*b)
781 -- Exponentiation with a cache for the most common numbers.
782 minExpt, maxExpt :: Int
786 expt :: Integer -> Int -> Integer
788 if base == 2 && n >= minExpt && n <= maxExpt then
793 expts :: Array Int Integer
794 expts = array (minExpt,maxExpt) [(n,2^n) | n <- [minExpt .. maxExpt]]
796 -- Compute the (floor of the) log of i in base b.
797 -- Simplest way would be just divide i by b until it's smaller then b, but that would
798 -- be very slow! We are just slightly more clever.
799 integerLogBase :: Integer -> Integer -> Int
802 | otherwise = doDiv (i `div` (b^l)) l
804 -- Try squaring the base first to cut down the number of divisions.
805 l = 2 * integerLogBase (b*b) i
807 doDiv :: Integer -> Int -> Int
810 | otherwise = doDiv (x `div` b) (y+1)
815 %*********************************************************
817 \subsection{Floating point numeric primops}
819 %*********************************************************
821 Definitions of the boxed PrimOps; these will be
822 used in the case of partial applications, etc.
825 plusFloat, minusFloat, timesFloat, divideFloat :: Float -> Float -> Float
826 plusFloat (F# x) (F# y) = F# (plusFloat# x y)
827 minusFloat (F# x) (F# y) = F# (minusFloat# x y)
828 timesFloat (F# x) (F# y) = F# (timesFloat# x y)
829 divideFloat (F# x) (F# y) = F# (divideFloat# x y)
831 negateFloat :: Float -> Float
832 negateFloat (F# x) = F# (negateFloat# x)
834 gtFloat, geFloat, eqFloat, neFloat, ltFloat, leFloat :: Float -> Float -> Bool
835 gtFloat (F# x) (F# y) = gtFloat# x y
836 geFloat (F# x) (F# y) = geFloat# x y
837 eqFloat (F# x) (F# y) = eqFloat# x y
838 neFloat (F# x) (F# y) = neFloat# x y
839 ltFloat (F# x) (F# y) = ltFloat# x y
840 leFloat (F# x) (F# y) = leFloat# x y
842 float2Int :: Float -> Int
843 float2Int (F# x) = I# (float2Int# x)
845 int2Float :: Int -> Float
846 int2Float (I# x) = F# (int2Float# x)
848 expFloat, logFloat, sqrtFloat :: Float -> Float
849 sinFloat, cosFloat, tanFloat :: Float -> Float
850 asinFloat, acosFloat, atanFloat :: Float -> Float
851 sinhFloat, coshFloat, tanhFloat :: Float -> Float
852 expFloat (F# x) = F# (expFloat# x)
853 logFloat (F# x) = F# (logFloat# x)
854 sqrtFloat (F# x) = F# (sqrtFloat# x)
855 sinFloat (F# x) = F# (sinFloat# x)
856 cosFloat (F# x) = F# (cosFloat# x)
857 tanFloat (F# x) = F# (tanFloat# x)
858 asinFloat (F# x) = F# (asinFloat# x)
859 acosFloat (F# x) = F# (acosFloat# x)
860 atanFloat (F# x) = F# (atanFloat# x)
861 sinhFloat (F# x) = F# (sinhFloat# x)
862 coshFloat (F# x) = F# (coshFloat# x)
863 tanhFloat (F# x) = F# (tanhFloat# x)
865 powerFloat :: Float -> Float -> Float
866 powerFloat (F# x) (F# y) = F# (powerFloat# x y)
868 -- definitions of the boxed PrimOps; these will be
869 -- used in the case of partial applications, etc.
871 plusDouble, minusDouble, timesDouble, divideDouble :: Double -> Double -> Double
872 plusDouble (D# x) (D# y) = D# (x +## y)
873 minusDouble (D# x) (D# y) = D# (x -## y)
874 timesDouble (D# x) (D# y) = D# (x *## y)
875 divideDouble (D# x) (D# y) = D# (x /## y)
877 negateDouble :: Double -> Double
878 negateDouble (D# x) = D# (negateDouble# x)
880 gtDouble, geDouble, eqDouble, neDouble, leDouble, ltDouble :: Double -> Double -> Bool
881 gtDouble (D# x) (D# y) = x >## y
882 geDouble (D# x) (D# y) = x >=## y
883 eqDouble (D# x) (D# y) = x ==## y
884 neDouble (D# x) (D# y) = x /=## y
885 ltDouble (D# x) (D# y) = x <## y
886 leDouble (D# x) (D# y) = x <=## y
888 double2Int :: Double -> Int
889 double2Int (D# x) = I# (double2Int# x)
891 int2Double :: Int -> Double
892 int2Double (I# x) = D# (int2Double# x)
894 double2Float :: Double -> Float
895 double2Float (D# x) = F# (double2Float# x)
897 float2Double :: Float -> Double
898 float2Double (F# x) = D# (float2Double# x)
900 expDouble, logDouble, sqrtDouble :: Double -> Double
901 sinDouble, cosDouble, tanDouble :: Double -> Double
902 asinDouble, acosDouble, atanDouble :: Double -> Double
903 sinhDouble, coshDouble, tanhDouble :: Double -> Double
904 expDouble (D# x) = D# (expDouble# x)
905 logDouble (D# x) = D# (logDouble# x)
906 sqrtDouble (D# x) = D# (sqrtDouble# x)
907 sinDouble (D# x) = D# (sinDouble# x)
908 cosDouble (D# x) = D# (cosDouble# x)
909 tanDouble (D# x) = D# (tanDouble# x)
910 asinDouble (D# x) = D# (asinDouble# x)
911 acosDouble (D# x) = D# (acosDouble# x)
912 atanDouble (D# x) = D# (atanDouble# x)
913 sinhDouble (D# x) = D# (sinhDouble# x)
914 coshDouble (D# x) = D# (coshDouble# x)
915 tanhDouble (D# x) = D# (tanhDouble# x)
917 powerDouble :: Double -> Double -> Double
918 powerDouble (D# x) (D# y) = D# (x **## y)
922 foreign import ccall unsafe "__encodeFloat"
923 encodeFloat# :: Int# -> ByteArray# -> Int -> Float
924 foreign import ccall unsafe "__int_encodeFloat"
925 int_encodeFloat# :: Int# -> Int -> Float
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 "__encodeDouble"
935 encodeDouble# :: Int# -> ByteArray# -> Int -> Double
936 foreign import ccall unsafe "__int_encodeDouble"
937 int_encodeDouble# :: Int# -> Int -> Double
939 foreign import ccall unsafe "isDoubleNaN" isDoubleNaN :: Double -> Int
940 foreign import ccall unsafe "isDoubleInfinite" isDoubleInfinite :: Double -> Int
941 foreign import ccall unsafe "isDoubleDenormalized" isDoubleDenormalized :: Double -> Int
942 foreign import ccall unsafe "isDoubleNegativeZero" isDoubleNegativeZero :: Double -> Int
945 %*********************************************************
947 \subsection{Coercion rules}
949 %*********************************************************
953 "fromIntegral/Int->Float" fromIntegral = int2Float
954 "fromIntegral/Int->Double" fromIntegral = int2Double
955 "realToFrac/Float->Float" realToFrac = id :: Float -> Float
956 "realToFrac/Float->Double" realToFrac = float2Double
957 "realToFrac/Double->Float" realToFrac = double2Float
958 "realToFrac/Double->Double" realToFrac = id :: Double -> Double