X-Git-Url: http://git.megacz.com/?a=blobdiff_plain;f=ghc%2Flib%2Fghc%2FPrelNum.lhs;fp=ghc%2Flib%2Fghc%2FPrelNum.lhs;h=0000000000000000000000000000000000000000;hb=28139aea50376444d56f43f0914291348a51a7e7;hp=3c1e4fee47c82ba1a07bdaf529e27b1e1dd415a0;hpb=98a1ebecb6d22d793b1d9f8e1d24ecbb5a2d130f;p=ghc-hetmet.git diff --git a/ghc/lib/ghc/PrelNum.lhs b/ghc/lib/ghc/PrelNum.lhs deleted file mode 100644 index 3c1e4fe..0000000 --- a/ghc/lib/ghc/PrelNum.lhs +++ /dev/null @@ -1,1265 +0,0 @@ -% -% (c) The AQUA Project, Glasgow University, 1994-1996 -% - -\section[PrelNum]{Module @PrelNum@} - -Numeric part of the prelude. - -It's rather big! - -\begin{code} -{-# OPTIONS -fno-implicit-prelude -#include "cbits/floatExtreme.h" #-} -{-# OPTIONS -H20m #-} - -#include "../includes/ieee-flpt.h" - -\end{code} - -\begin{code} -module PrelNum where - -import PrelBase -import GHC -import {-# SOURCE #-} GHCerr ( error ) -import PrelList -import PrelMaybe - -import ArrBase ( Array, array, (!) ) -import Unsafe ( unsafePerformIO ) -import Ix ( Ix(..) ) -import CCall () -- we need the definitions of CCallable and CReturnable - -- for the _ccall_s herein. - - -infixr 8 ^, ^^, ** -infixl 7 /, %, `quot`, `rem`, `div`, `mod` -\end{code} - - -%********************************************************* -%* * -\subsection{Standard numeric classes} -%* * -%********************************************************* - -\begin{code} -class (Num a, Ord a) => Real a where - toRational :: a -> Rational - -class (Real a, Enum a) => Integral a where - quot, rem, div, mod :: a -> a -> a - quotRem, divMod :: a -> a -> (a,a) - toInteger :: a -> Integer - toInt :: a -> Int -- partain: Glasgow extension - - n `quot` d = q where (q,r) = quotRem n d - n `rem` d = r where (q,r) = quotRem n d - n `div` d = q where (q,r) = divMod n d - n `mod` d = r where (q,r) = divMod n d - divMod n d = if signum r == negate (signum d) then (q-1, r+d) else qr - where qr@(q,r) = quotRem n d - -class (Num a) => Fractional a where - (/) :: a -> a -> a - recip :: a -> a - fromRational :: Rational -> a - - recip x = 1 / x - -class (Fractional a) => Floating a where - pi :: a - exp, log, sqrt :: a -> a - (**), logBase :: a -> a -> a - sin, cos, tan :: a -> a - asin, acos, atan :: a -> a - sinh, cosh, tanh :: a -> a - asinh, acosh, atanh :: a -> a - - x ** y = exp (log x * y) - logBase x y = log y / log x - sqrt x = x ** 0.5 - tan x = sin x / cos x - tanh x = sinh x / cosh x - -class (Real a, Fractional a) => RealFrac a where - properFraction :: (Integral b) => a -> (b,a) - truncate, round :: (Integral b) => a -> b - ceiling, floor :: (Integral b) => a -> b - - truncate x = m where (m,_) = properFraction x - - round x = let (n,r) = properFraction x - m = if r < 0 then n - 1 else n + 1 - in case signum (abs r - 0.5) of - -1 -> n - 0 -> if even n then n else m - 1 -> m - - ceiling x = if r > 0 then n + 1 else n - where (n,r) = properFraction x - - floor x = if r < 0 then n - 1 else n - where (n,r) = properFraction x - -class (RealFrac a, Floating a) => RealFloat a where - floatRadix :: a -> Integer - floatDigits :: a -> Int - floatRange :: a -> (Int,Int) - decodeFloat :: a -> (Integer,Int) - encodeFloat :: Integer -> Int -> a - exponent :: a -> Int - significand :: a -> a - scaleFloat :: Int -> a -> a - isNaN, isInfinite, isDenormalized, isNegativeZero, isIEEE - :: a -> Bool - - exponent x = if m == 0 then 0 else n + floatDigits x - where (m,n) = decodeFloat x - - significand x = encodeFloat m (negate (floatDigits x)) - where (m,_) = decodeFloat x - - scaleFloat k x = encodeFloat m (n+k) - where (m,n) = decodeFloat x -\end{code} - -%********************************************************* -%* * -\subsection{Overloaded numeric functions} -%* * -%********************************************************* - -\begin{code} -even, odd :: (Integral a) => a -> Bool -even n = n `rem` 2 == 0 -odd = not . even - -{-# GENERATE_SPECS gcd a{Int#,Int,Integer} #-} -gcd :: (Integral a) => a -> a -> a -gcd 0 0 = error "Prelude.gcd: gcd 0 0 is undefined" -gcd x y = gcd' (abs x) (abs y) - where gcd' x 0 = x - gcd' x y = gcd' y (x `rem` y) - -{-# GENERATE_SPECS lcm a{Int#,Int,Integer} #-} -lcm :: (Integral a) => a -> a -> a -lcm _ 0 = 0 -lcm 0 _ = 0 -lcm x y = abs ((x `quot` (gcd x y)) * y) - -(^) :: (Num a, Integral b) => a -> b -> a -x ^ 0 = 1 -x ^ n | n > 0 = f x (n-1) x - where f _ 0 y = y - f x n y = g x n where - g x n | even n = g (x*x) (n `quot` 2) - | otherwise = f x (n-1) (x*y) -_ ^ _ = error "Prelude.^: negative exponent" - -(^^) :: (Fractional a, Integral b) => a -> b -> a -x ^^ n = if n >= 0 then x^n else recip (x^(negate n)) - -fromIntegral :: (Integral a, Num b) => a -> b -fromIntegral = fromInteger . toInteger - -fromRealFrac :: (RealFrac a, Fractional b) => a -> b -fromRealFrac = fromRational . toRational - -atan2 :: (RealFloat a) => a -> a -> a -atan2 y x = case (signum y, signum x) of - ( 0, 1) -> 0 - ( 1, 0) -> pi/2 - ( 0,-1) -> pi - (-1, 0) -> (negate pi)/2 - ( _, 1) -> atan (y/x) - ( _,-1) -> atan (y/x) + pi - ( 0, 0) -> error "Prelude.atan2: atan2 of origin" -\end{code} - - -%********************************************************* -%* * -\subsection{Instances for @Int@} -%* * -%********************************************************* - -\begin{code} -instance Real Int where - toRational x = toInteger x % 1 - -instance Integral Int where - a@(I# _) `quotRem` b@(I# _) = (a `quotInt` b, a `remInt` b) - -- OK, so I made it a little stricter. Shoot me. (WDP 94/10) - - -- Following chks for zero divisor are non-standard (WDP) - a `quot` b = if b /= 0 - then a `quotInt` b - else error "Integral.Int.quot{PreludeCore}: divide by 0\n" - a `rem` b = if b /= 0 - then a `remInt` b - else error "Integral.Int.rem{PreludeCore}: divide by 0\n" - - x `div` y = if x > 0 && y < 0 then quotInt (x-y-1) y - else if x < 0 && y > 0 then quotInt (x-y+1) y - else quotInt x y - x `mod` y = if x > 0 && y < 0 || x < 0 && y > 0 then - if r/=0 then r+y else 0 - else - r - where r = remInt x y - - divMod x@(I# _) y@(I# _) = (x `div` y, x `mod` y) - -- Stricter. Sorry if you don't like it. (WDP 94/10) - ---OLD: even x = eqInt (x `mod` 2) 0 ---OLD: odd x = neInt (x `mod` 2) 0 - - toInteger (I# n#) = int2Integer# n# -- give back a full-blown Integer - toInt x = x - -\end{code} - - -%********************************************************* -%* * -\subsection{Type @Integer@} -%* * -%********************************************************* - -These types are used to return from integer primops - -\begin{code} -data Return2GMPs = Return2GMPs Int# Int# ByteArray# Int# Int# ByteArray# -data ReturnIntAndGMP = ReturnIntAndGMP Int# Int# Int# ByteArray# -\end{code} - -Instances - -\begin{code} -instance Eq Integer where - (J# a1 s1 d1) == (J# a2 s2 d2) - = (cmpInteger# a1 s1 d1 a2 s2 d2) ==# 0# - - (J# a1 s1 d1) /= (J# a2 s2 d2) - = (cmpInteger# a1 s1 d1 a2 s2 d2) /=# 0# - -instance Ord Integer where - (J# a1 s1 d1) <= (J# a2 s2 d2) - = (cmpInteger# a1 s1 d1 a2 s2 d2) <=# 0# - - (J# a1 s1 d1) < (J# a2 s2 d2) - = (cmpInteger# a1 s1 d1 a2 s2 d2) <# 0# - - (J# a1 s1 d1) >= (J# a2 s2 d2) - = (cmpInteger# a1 s1 d1 a2 s2 d2) >=# 0# - - (J# a1 s1 d1) > (J# a2 s2 d2) - = (cmpInteger# a1 s1 d1 a2 s2 d2) ># 0# - - x@(J# a1 s1 d1) `max` y@(J# a2 s2 d2) - = if ((cmpInteger# a1 s1 d1 a2 s2 d2) ># 0#) then x else y - - x@(J# a1 s1 d1) `min` y@(J# a2 s2 d2) - = if ((cmpInteger# a1 s1 d1 a2 s2 d2) <# 0#) then x else y - - compare (J# a1 s1 d1) (J# a2 s2 d2) - = case cmpInteger# a1 s1 d1 a2 s2 d2 of { res# -> - if res# <# 0# then LT else - if res# ># 0# then GT else EQ - } - -instance Num Integer where - (+) (J# a1 s1 d1) (J# a2 s2 d2) - = plusInteger# a1 s1 d1 a2 s2 d2 - - (-) (J# a1 s1 d1) (J# a2 s2 d2) - = minusInteger# a1 s1 d1 a2 s2 d2 - - negate (J# a s d) = negateInteger# a s d - - (*) (J# a1 s1 d1) (J# a2 s2 d2) - = timesInteger# a1 s1 d1 a2 s2 d2 - - -- ORIG: abs n = if n >= 0 then n else -n - - abs n@(J# a1 s1 d1) - = case 0 of { J# a2 s2 d2 -> - if (cmpInteger# a1 s1 d1 a2 s2 d2) >=# 0# - then n - else negateInteger# a1 s1 d1 - } - - signum n@(J# a1 s1 d1) - = case 0 of { J# a2 s2 d2 -> - let - cmp = cmpInteger# a1 s1 d1 a2 s2 d2 - in - if cmp ># 0# then 1 - else if cmp ==# 0# then 0 - else (negate 1) - } - - fromInteger x = x - - fromInt (I# n#) = int2Integer# n# -- gives back a full-blown Integer - -instance Real Integer where - toRational x = x % 1 - -instance Integral Integer where - quotRem (J# a1 s1 d1) (J# a2 s2 d2) - = case (quotRemInteger# a1 s1 d1 a2 s2 d2) of - Return2GMPs a3 s3 d3 a4 s4 d4 - -> (J# a3 s3 d3, J# a4 s4 d4) - -{- USING THE UNDERLYING "GMP" CODE IS DUBIOUS FOR NOW: - - divMod (J# a1 s1 d1) (J# a2 s2 d2) - = case (divModInteger# a1 s1 d1 a2 s2 d2) of - Return2GMPs a3 s3 d3 a4 s4 d4 - -> (J# a3 s3 d3, J# a4 s4 d4) --} - toInteger n = n - toInt (J# a s d) = case (integer2Int# a s d) of { n# -> I# n# } - - -- the rest are identical to the report default methods; - -- you get slightly better code if you let the compiler - -- see them right here: - n `quot` d = if d /= 0 then q else - error "Integral.Integer.quot{PreludeCore}: divide by 0\n" - where (q,r) = quotRem n d - n `rem` d = if d /= 0 then r else - error "Integral.Integer.quot{PreludeCore}: divide by 0\n" - where (q,r) = quotRem n d - n `div` d = q where (q,r) = divMod n d - n `mod` d = r where (q,r) = divMod n d - - divMod n d = case (quotRem n d) of { qr@(q,r) -> - if signum r == negate (signum d) then (q - 1, r+d) else qr } - -- Case-ified by WDP 94/10 - -instance Enum Integer where - toEnum n = toInteger n - fromEnum n = toInt n - enumFrom n = n : enumFrom (n + 1) - enumFromThen m n = en' m (n - m) - where en' m n = m : en' (m + n) n - enumFromTo n m = takeWhile (<= m) (enumFrom n) - enumFromThenTo n m p = takeWhile (if m >= n then (<= p) else (>= p)) - (enumFromThen n m) - -instance Show Integer where - showsPrec x = showSignedInteger x - showList = showList__ (showsPrec 0) - -instance Ix Integer where - range (m,n) = [m..n] - index b@(m,n) i - | inRange b i = fromInteger (i - m) - | otherwise = error "Integer.index: Index out of range." - inRange (m,n) i = m <= i && i <= n - -integer_0, integer_1, integer_2, integer_m1 :: Integer -integer_0 = int2Integer# 0# -integer_1 = int2Integer# 1# -integer_2 = int2Integer# 2# -integer_m1 = int2Integer# (negateInt# 1#) -\end{code} - - -%********************************************************* -%* * -\subsection{Type @Float@} -%* * -%********************************************************* - -\begin{code} -instance Eq Float where - (F# x) == (F# y) = x `eqFloat#` y - -instance Ord Float where - (F# x) `compare` (F# y) | x `ltFloat#` y = LT - | x `eqFloat#` y = EQ - | otherwise = GT - - (F# x) < (F# y) = x `ltFloat#` y - (F# x) <= (F# y) = x `leFloat#` y - (F# x) >= (F# y) = x `geFloat#` y - (F# x) > (F# y) = x `gtFloat#` y - -instance Num Float where - (+) x y = plusFloat x y - (-) x y = minusFloat x y - negate x = negateFloat x - (*) x y = timesFloat x y - abs x | x >= 0.0 = x - | otherwise = negateFloat x - signum x | x == 0.0 = 0 - | x > 0.0 = 1 - | otherwise = negate 1 - fromInteger n = encodeFloat n 0 - fromInt i = int2Float i - -instance Real Float where - toRational x = (m%1)*(b%1)^^n - where (m,n) = decodeFloat x - b = floatRadix x - -instance Fractional Float where - (/) x y = divideFloat x y - fromRational x = fromRat x - recip x = 1.0 / x - -instance Floating Float where - pi = 3.141592653589793238 - exp x = expFloat x - log x = logFloat x - sqrt x = sqrtFloat x - sin x = sinFloat x - cos x = cosFloat x - tan x = tanFloat x - asin x = asinFloat x - acos x = acosFloat x - atan x = atanFloat x - sinh x = sinhFloat x - cosh x = coshFloat x - tanh x = tanhFloat x - (**) x y = powerFloat x y - logBase x y = log y / log x - - asinh x = log (x + sqrt (1.0+x*x)) - acosh x = log (x + (x+1.0) * sqrt ((x-1.0)/(x+1.0))) - atanh x = log ((x+1.0) / sqrt (1.0-x*x)) - -instance RealFrac Float where - - {-# SPECIALIZE properFraction :: Float -> (Int, Float) #-} - {-# SPECIALIZE truncate :: Float -> Int #-} - {-# SPECIALIZE round :: Float -> Int #-} - {-# SPECIALIZE ceiling :: Float -> Int #-} - {-# SPECIALIZE floor :: Float -> Int #-} - - {-# SPECIALIZE properFraction :: Float -> (Integer, Float) #-} - {-# SPECIALIZE truncate :: Float -> Integer #-} - {-# SPECIALIZE round :: Float -> Integer #-} - {-# SPECIALIZE ceiling :: Float -> Integer #-} - {-# SPECIALIZE floor :: Float -> Integer #-} - - properFraction x - = case (decodeFloat x) of { (m,n) -> - let b = floatRadix x in - if n >= 0 then - (fromInteger m * fromInteger b ^ n, 0.0) - else - case (quotRem m (b^(negate n))) of { (w,r) -> - (fromInteger w, encodeFloat r n) - } - } - - truncate x = case properFraction x of - (n,_) -> n - - round x = case properFraction x of - (n,r) -> let - m = if r < 0.0 then n - 1 else n + 1 - half_down = abs r - 0.5 - in - case (compare half_down 0.0) of - LT -> n - EQ -> if even n then n else m - GT -> m - - ceiling x = case properFraction x of - (n,r) -> if r > 0.0 then n + 1 else n - - floor x = case properFraction x of - (n,r) -> if r < 0.0 then n - 1 else n - -instance RealFloat Float where - floatRadix _ = FLT_RADIX -- from float.h - floatDigits _ = FLT_MANT_DIG -- ditto - floatRange _ = (FLT_MIN_EXP, FLT_MAX_EXP) -- ditto - - decodeFloat (F# f#) - = case decodeFloat# f# of - ReturnIntAndGMP exp# a# s# d# -> - (J# a# s# d#, I# exp#) - - encodeFloat (J# a# s# d#) (I# e#) - = case encodeFloat# a# s# d# e# of { flt# -> F# flt# } - - exponent x = case decodeFloat x of - (m,n) -> if m == 0 then 0 else n + floatDigits x - - significand x = case decodeFloat x of - (m,_) -> encodeFloat m (negate (floatDigits x)) - - scaleFloat k x = case decodeFloat x of - (m,n) -> encodeFloat m (n+k) - isNaN x = - (0::Int) /= unsafePerformIO (_ccall_ isFloatNaN x) {- a _pure_function! -} - isInfinite x = - (0::Int) /= unsafePerformIO (_ccall_ isFloatInfinite x) {- ditto! -} - isDenormalized x = - (0::Int) /= unsafePerformIO (_ccall_ isFloatDenormalized x) -- .. - isNegativeZero x = - (0::Int) /= unsafePerformIO (_ccall_ isFloatNegativeZero x) -- ... - isIEEE x = True - -instance Show Float where - showsPrec x = showSigned showFloat x - showList = showList__ (showsPrec 0) -\end{code} - -%********************************************************* -%* * -\subsection{Type @Double@} -%* * -%********************************************************* - -\begin{code} -instance Eq Double where - (D# x) == (D# y) = x ==## y - -instance Ord Double where - (D# x) `compare` (D# y) | x <## y = LT - | x ==## y = EQ - | otherwise = GT - - (D# x) < (D# y) = x <## y - (D# x) <= (D# y) = x <=## y - (D# x) >= (D# y) = x >=## y - (D# x) > (D# y) = x >## y - -instance Num Double where - (+) x y = plusDouble x y - (-) x y = minusDouble x y - negate x = negateDouble x - (*) x y = timesDouble x y - abs x | x >= 0.0 = x - | otherwise = negateDouble x - signum x | x == 0.0 = 0 - | x > 0.0 = 1 - | otherwise = negate 1 - fromInteger n = encodeFloat n 0 - fromInt (I# n#) = case (int2Double# n#) of { d# -> D# d# } - -instance Real Double where - toRational x = (m%1)*(b%1)^^n - where (m,n) = decodeFloat x - b = floatRadix x - -instance Fractional Double where - (/) x y = divideDouble x y - fromRational x = fromRat x - recip x = 1.0 / x - -instance Floating Double where - pi = 3.141592653589793238 - exp x = expDouble x - log x = logDouble x - sqrt x = sqrtDouble x - sin x = sinDouble x - cos x = cosDouble x - tan x = tanDouble x - asin x = asinDouble x - acos x = acosDouble x - atan x = atanDouble x - sinh x = sinhDouble x - cosh x = coshDouble x - tanh x = tanhDouble x - (**) x y = powerDouble x y - logBase x y = log y / log x - - asinh x = log (x + sqrt (1.0+x*x)) - acosh x = log (x + (x+1.0) * sqrt ((x-1.0)/(x+1.0))) - atanh x = log ((x+1.0) / sqrt (1.0-x*x)) - -instance RealFrac Double where - - {-# SPECIALIZE properFraction :: Double -> (Int, Double) #-} - {-# SPECIALIZE truncate :: Double -> Int #-} - {-# SPECIALIZE round :: Double -> Int #-} - {-# SPECIALIZE ceiling :: Double -> Int #-} - {-# SPECIALIZE floor :: Double -> Int #-} - - {-# SPECIALIZE properFraction :: Double -> (Integer, Double) #-} - {-# SPECIALIZE truncate :: Double -> Integer #-} - {-# SPECIALIZE round :: Double -> Integer #-} - {-# SPECIALIZE ceiling :: Double -> Integer #-} - {-# SPECIALIZE floor :: Double -> Integer #-} - -#if defined(__UNBOXED_INSTANCES__) - {-# SPECIALIZE properFraction :: Double -> (Int#, Double) #-} - {-# SPECIALIZE truncate :: Double -> Int# #-} - {-# SPECIALIZE round :: Double -> Int# #-} - {-# SPECIALIZE ceiling :: Double -> Int# #-} - {-# SPECIALIZE floor :: Double -> Int# #-} -#endif - - properFraction x - = case (decodeFloat x) of { (m,n) -> - let b = floatRadix x in - if n >= 0 then - (fromInteger m * fromInteger b ^ n, 0.0) - else - case (quotRem m (b^(negate n))) of { (w,r) -> - (fromInteger w, encodeFloat r n) - } - } - - truncate x = case properFraction x of - (n,_) -> n - - round x = case properFraction x of - (n,r) -> let - m = if r < 0.0 then n - 1 else n + 1 - half_down = abs r - 0.5 - in - case (compare half_down 0.0) of - LT -> n - EQ -> if even n then n else m - GT -> m - - ceiling x = case properFraction x of - (n,r) -> if r > 0.0 then n + 1 else n - - floor x = case properFraction x of - (n,r) -> if r < 0.0 then n - 1 else n - -instance RealFloat Double where - floatRadix _ = FLT_RADIX -- from float.h - floatDigits _ = DBL_MANT_DIG -- ditto - floatRange _ = (DBL_MIN_EXP, DBL_MAX_EXP) -- ditto - - decodeFloat (D# d#) - = case decodeDouble# d# of - ReturnIntAndGMP exp# a# s# d# -> - (J# a# s# d#, I# exp#) - - encodeFloat (J# a# s# d#) (I# e#) - = case encodeDouble# a# s# d# e# of { dbl# -> D# dbl# } - - exponent x = case decodeFloat x of - (m,n) -> if m == 0 then 0 else n + floatDigits x - - significand x = case decodeFloat x of - (m,_) -> encodeFloat m (negate (floatDigits x)) - - scaleFloat k x = case decodeFloat x of - (m,n) -> encodeFloat m (n+k) - isNaN x = - (0::Int) /= unsafePerformIO (_ccall_ isDoubleNaN x) {- a _pure_function! -} - isInfinite x = - (0::Int) /= unsafePerformIO (_ccall_ isDoubleInfinite x) {- ditto -} - isDenormalized x = - (0::Int) /= unsafePerformIO (_ccall_ isDoubleDenormalized x) -- .. - isNegativeZero x = - (0::Int) /= unsafePerformIO (_ccall_ isDoubleNegativeZero x) -- ... - isIEEE x = True - -instance Show Double where - showsPrec x = showSigned showFloat x - showList = showList__ (showsPrec 0) -\end{code} - - -%********************************************************* -%* * -\subsection{Common code for @Float@ and @Double@} -%* * -%********************************************************* - -The @Enum@ instances for Floats and Doubles are slightly unusual. -The @toEnum@ function truncates numbers to Int. The definitions -of @enumFrom@ and @enumFromThen@ allow floats to be used in arithmetic -series: [0,0.1 .. 1.0]. However, roundoff errors make these somewhat -dubious. This example may have either 10 or 11 elements, depending on -how 0.1 is represented. - -NOTE: The instances for Float and Double do not make use of the default -methods for @enumFromTo@ and @enumFromThenTo@, as these rely on there being -a `non-lossy' conversion to and from Ints. Instead we make use of the -1.2 default methods (back in the days when Enum had Ord as a superclass) -for these (@numericEnumFromTo@ and @numericEnumFromThenTo@ below.) - -\begin{code} -instance Enum Float where - toEnum = fromIntegral - fromEnum = fromInteger . truncate -- may overflow - enumFrom = numericEnumFrom - enumFromThen = numericEnumFromThen - enumFromThenTo = numericEnumFromThenTo - -instance Enum Double where - toEnum = fromIntegral - fromEnum = fromInteger . truncate -- may overflow - enumFrom = numericEnumFrom - enumFromThen = numericEnumFromThen - enumFromThenTo = numericEnumFromThenTo - -numericEnumFrom :: (Real a) => a -> [a] -numericEnumFromThen :: (Real a) => a -> a -> [a] -numericEnumFromThenTo :: (Real a) => a -> a -> a -> [a] -numericEnumFrom = iterate (+1) -numericEnumFromThen n m = iterate (+(m-n)) n -numericEnumFromThenTo n m p = takeWhile (if m >= n then (<= p) else (>= p)) - (numericEnumFromThen n m) -\end{code} - - -%********************************************************* -%* * -\subsection{The @Ratio@ and @Rational@ types} -%* * -%********************************************************* - -\begin{code} -data (Eval a, Integral a) => Ratio a = !a :% !a deriving (Eq) -type Rational = Ratio Integer -\end{code} - -\begin{code} -(%) :: (Integral a) => a -> a -> Ratio a -numerator, denominator :: (Integral a) => Ratio a -> a -approxRational :: (RealFrac a) => a -> a -> Rational - -\end{code} - -\tr{reduce} is a subsidiary function used only in this module . -It normalises a ratio by dividing both numerator and denominator by -their greatest common divisor. - -\begin{code} -reduce x 0 = error "{Ratio.%}: zero denominator" -reduce x y = (x `quot` d) :% (y `quot` d) - where d = gcd x y -\end{code} - -\begin{code} -x % y = reduce (x * signum y) (abs y) - -numerator (x:%y) = x - -denominator (x:%y) = y -\end{code} - - -@approxRational@, applied to two real fractional numbers x and epsilon, -returns the simplest rational number within epsilon of x. A rational -number n%d in reduced form is said to be simpler than another n'%d' if -abs n <= abs n' && d <= d'. Any real interval contains a unique -simplest rational; here, for simplicity, we assume a closed rational -interval. If such an interval includes at least one whole number, then -the simplest rational is the absolutely least whole number. Otherwise, -the bounds are of the form q%1 + r%d and q%1 + r'%d', where abs r < d -and abs r' < d', and the simplest rational is q%1 + the reciprocal of -the simplest rational between d'%r' and d%r. - -\begin{code} -approxRational x eps = simplest (x-eps) (x+eps) - where simplest x y | y < x = simplest y x - | x == y = xr - | x > 0 = simplest' n d n' d' - | y < 0 = - simplest' (-n') d' (-n) d - | otherwise = 0 :% 1 - where xr = toRational x - n = numerator xr - d = denominator xr - nd' = toRational y - n' = numerator nd' - d' = denominator nd' - - simplest' n d n' d' -- assumes 0 < n%d < n'%d' - | r == 0 = q :% 1 - | q /= q' = (q+1) :% 1 - | otherwise = (q*n''+d'') :% n'' - where (q,r) = quotRem n d - (q',r') = quotRem n' d' - nd'' = simplest' d' r' d r - n'' = numerator nd'' - d'' = denominator nd'' -\end{code} - - -\begin{code} -instance (Integral a) => Ord (Ratio a) where - (x:%y) <= (x':%y') = x * y' <= x' * y - (x:%y) < (x':%y') = x * y' < x' * y - -instance (Integral a) => Num (Ratio a) where - (x:%y) + (x':%y') = reduce (x*y' + x'*y) (y*y') - (x:%y) - (x':%y') = reduce (x*y' - x'*y) (y*y') - (x:%y) * (x':%y') = reduce (x * x') (y * y') - negate (x:%y) = (-x) :% y - abs (x:%y) = abs x :% y - signum (x:%y) = signum x :% 1 - fromInteger x = fromInteger x :% 1 - -instance (Integral a) => Real (Ratio a) where - toRational (x:%y) = toInteger x :% toInteger y - -instance (Integral a) => Fractional (Ratio a) where - (x:%y) / (x':%y') = (x*y') % (y*x') - recip (x:%y) = if x < 0 then (-y) :% (-x) else y :% x - fromRational (x:%y) = fromInteger x :% fromInteger y - -instance (Integral a) => RealFrac (Ratio a) where - properFraction (x:%y) = (fromIntegral q, r:%y) - where (q,r) = quotRem x y - -instance (Integral a) => Enum (Ratio a) where - enumFrom = iterate ((+)1) - enumFromThen n m = iterate ((+)(m-n)) n - toEnum n = fromIntegral n :% 1 - fromEnum = fromInteger . truncate - -ratio_prec :: Int -ratio_prec = 7 - -instance (Integral a) => Show (Ratio a) where - showsPrec p (x:%y) = showParen (p > ratio_prec) - (shows x . showString " % " . shows y) -\end{code} - -\begin{code} ---Exported from std library Numeric, defined here to ---avoid mut. rec. between PrelNum and Numeric. -showSigned :: (Real a) => (a -> ShowS) -> Int -> a -> ShowS -showSigned showPos p x = if x < 0 then showParen (p > 6) - (showChar '-' . showPos (-x)) - else showPos x - -showSignedInteger :: Int -> Integer -> ShowS -showSignedInteger p n r - = -- from HBC version; support code follows - if n < 0 && p > 6 then '(':jtos n++(')':r) else jtos n ++ r - -jtos :: Integer -> String -jtos n - = if n < 0 then - '-' : jtos' (-n) [] - else - jtos' n [] - -jtos' :: Integer -> String -> String -jtos' n cs - = if n < 10 then - chr (fromInteger (n + ord_0)) : cs - else - jtos' (n `quot` 10) (chr (fromInteger (n `rem` 10 + ord_0)) : cs) - -showFloat x = showString (formatRealFloat FFGeneric Nothing x) - --- These are the format types. This type is not exported. - -data FFFormat = FFExponent | FFFixed | FFGeneric --no need: deriving (Eq, Ord, Show) - -formatRealFloat :: (RealFloat a) => FFFormat -> Maybe Int -> a -> String -formatRealFloat fmt decs x = s - where - base = 10 - s = if isNaN x - then "NaN" - else - if isInfinite x then - if x < 0 then "-Infinity" else "Infinity" - else - if x < 0 || isNegativeZero x then - '-':doFmt fmt (floatToDigits (toInteger base) (-x)) - else - doFmt fmt (floatToDigits (toInteger base) x) - - doFmt fmt (is, e) = - let ds = map intToDigit is in - case fmt of - FFGeneric -> - doFmt (if e <0 || e > 7 then FFExponent else FFFixed) - (is,e) - FFExponent -> - case decs of - Nothing -> - let e' = if e==0 then 0 else e-1 in - (case ds of - [d] -> d : ".0e" - (d:ds) -> d : '.' : ds ++ "e") ++ show e' - Just dec -> - let dec' = max dec 1 in - case is of - [0] -> '0':'.':take dec' (repeat '0') ++ "e0" - _ -> - let - (ei,is') = roundTo base (dec'+1) is - d:ds = map intToDigit (if ei > 0 then init is' else is') - in - d:'.':ds ++ 'e':show (e-1+ei) - FFFixed -> - let - mk0 ls = case ls of { "" -> "0" ; _ -> ls} - in - case decs of - Nothing -> - let - f 0 s ds = mk0 (reverse s) ++ '.':mk0 ds - f n s "" = f (n-1) ('0':s) "" - f n s (d:ds) = f (n-1) (d:s) ds - in - f e "" ds - Just dec -> - let dec' = max dec 1 in - if e >= 0 then - let - (ei,is') = roundTo base (dec' + e) is - (ls,rs) = splitAt (e+ei) (map intToDigit is') - in - mk0 ls ++ (if null rs then "" else '.':rs) - else - let - (ei,is') = roundTo base dec' (replicate (-e) 0 ++ is) - d:ds = map intToDigit (if ei > 0 then is' else 0:is') - in - d : '.' : ds - - -roundTo :: Int -> Int -> [Int] -> (Int,[Int]) -roundTo base d is = - let - v = f d is - in - case v of - (0,is) -> v - (1,is) -> (1, 1:is) - where - b2 = base `div` 2 - - f n [] = (0, replicate n 0) - f 0 (i:_) = (if i>=b2 then 1 else 0, []) - f d (i:is) = - let - (c,ds) = f (d-1) is - i' = c + i - in - if i' == base then (1,0:ds) else (0,i':ds) - --- --- Based on "Printing Floating-Point Numbers Quickly and Accurately" --- by R.G. Burger and R.K. Dybvig in PLDI 96. --- This version uses a much slower logarithm estimator. It should be improved. - --- This function returns a list of digits (Ints in [0..base-1]) and an --- exponent. ---floatToDigits :: (RealFloat a) => Integer -> a -> ([Int], Int) -floatToDigits _ 0 = ([0], 0) -floatToDigits base x = - let - (f0, e0) = decodeFloat x - (minExp0, _) = floatRange x - p = floatDigits x - b = floatRadix x - minExp = minExp0 - p -- the real minimum exponent - -- Haskell requires that f be adjusted so denormalized numbers - -- will have an impossibly low exponent. Adjust for this. - (f, e) = - let n = minExp - e0 in - if n > 0 then (f0 `div` (b^n), e0+n) else (f0, e0) - (r, s, mUp, mDn) = - if e >= 0 then - let be = b^ e in - if f == b^(p-1) then - (f*be*b*2, 2*b, be*b, b) - else - (f*be*2, 2, be, be) - else - if e > minExp && f == b^(p-1) then - (f*b*2, b^(-e+1)*2, b, 1) - else - (f*2, b^(-e)*2, 1, 1) - k = - let - k0 = - if b == 2 && base == 10 then - -- logBase 10 2 is slightly bigger than 3/10 so - -- the following will err on the low side. Ignoring - -- the fraction will make it err even more. - -- Haskell promises that p-1 <= logBase b f < p. - (p - 1 + e0) * 3 `div` 10 - else - ceiling ((log (fromInteger (f+1)) + - fromInt e * log (fromInteger b)) / - fromInt e * log (fromInteger b)) - - fixup n = - if n >= 0 then - if r + mUp <= expt base n * s then n else fixup (n+1) - else - if expt base (-n) * (r + mUp) <= s then n else fixup (n+1) - in - fixup k0 - - gen ds rn sN mUpN mDnN = - let - (dn, rn') = (rn * base) `divMod` sN - mUpN' = mUpN * base - mDnN' = mDnN * base - in - case (rn' < mDnN', rn' + mUpN' > sN) of - (True, False) -> dn : ds - (False, True) -> dn+1 : ds - (True, True) -> if rn' * 2 < sN then dn : ds else dn+1 : ds - (False, False) -> gen (dn:ds) rn' sN mUpN' mDnN' - - rds = - if k >= 0 then - gen [] r (s * expt base k) mUp mDn - else - let bk = expt base (-k) in - gen [] (r * bk) s (mUp * bk) (mDn * bk) - in - (map toInt (reverse rds), k) - -\end{code} - -@showRational@ converts a Rational to a string that looks like a -floating point number, but without converting to any floating type -(because of the possible overflow). - -From/by Lennart, 94/09/26 - -\begin{code} -showRational :: Int -> Rational -> String -showRational n r = - if r == 0 then - "0.0" - else - let (r', e) = normalize r - in prR n r' e - -startExpExp = 4 :: Int - --- make sure 1 <= r < 10 -normalize :: Rational -> (Rational, Int) -normalize r = if r < 1 then - case norm startExpExp (1 / r) 0 of (r', e) -> (10 / r', -e-1) - else - norm startExpExp r 0 - where norm :: Int -> Rational -> Int -> (Rational, Int) - -- Invariant: r*10^e == original r - norm 0 r e = (r, e) - norm ee r e = - let n = 10^ee - tn = 10^n - in if r >= tn then norm ee (r/tn) (e+n) else norm (ee-1) r e - -drop0 "" = "" -drop0 (c:cs) = c : reverse (dropWhile (=='0') (reverse cs)) - -prR :: Int -> Rational -> Int -> String -prR n r e | r < 1 = prR n (r*10) (e-1) -- final adjustment -prR n r e | r >= 10 = prR n (r/10) (e+1) -prR n r e0 = - let s = show ((round (r * 10^n))::Integer) - e = e0+1 - in if e > 0 && e < 8 then - take e s ++ "." ++ drop0 (drop e s) - else if e <= 0 && e > -3 then - "0." ++ take (-e) (repeat '0') ++ drop0 s - else - head s : "."++ drop0 (tail s) ++ "e" ++ show e0 -\end{code} - - -[In response to a request for documentation of how fromRational works, -Joe Fasel writes:] A quite reasonable request! This code was added to -the Prelude just before the 1.2 release, when Lennart, working with an -early version of hbi, noticed that (read . show) was not the identity -for floating-point numbers. (There was a one-bit error about half the -time.) The original version of the conversion function was in fact -simply a floating-point divide, as you suggest above. The new version -is, I grant you, somewhat denser. - -Unfortunately, Joe's code doesn't work! Here's an example: - -main = putStr (shows (1.82173691287639817263897126389712638972163e-300::Double) "\n") - -This program prints - 0.0000000000000000 -instead of - 1.8217369128763981e-300 - -Lennart's code follows, and it works... - -\begin{pseudocode} -{-# GENERATE_SPECS fromRational__ a{Double#,Double} #-} -fromRat :: (RealFloat a) => Rational -> a -fromRat x = x' - where x' = f e - --- If the exponent of the nearest floating-point number to x --- is e, then the significand is the integer nearest xb^(-e), --- where b is the floating-point radix. We start with a good --- guess for e, and if it is correct, the exponent of the --- floating-point number we construct will again be e. If --- not, one more iteration is needed. - - f e = if e' == e then y else f e' - where y = encodeFloat (round (x * (1 % b)^^e)) e - (_,e') = decodeFloat y - b = floatRadix x' - --- We obtain a trial exponent by doing a floating-point --- division of x's numerator by its denominator. The --- result of this division may not itself be the ultimate --- result, because of an accumulation of three rounding --- errors. - - (s,e) = decodeFloat (fromInteger (numerator x) `asTypeOf` x' - / fromInteger (denominator x)) -\end{pseudocode} - -Now, here's Lennart's code. - -\begin{code} ---fromRat :: (RealFloat a) => Rational -> a -fromRat x = - if x == 0 then encodeFloat 0 0 -- Handle exceptional cases - else if x < 0 then - fromRat' (-x) -- first. - else fromRat' x - --- Conversion process: --- Scale the rational number by the RealFloat base until --- it lies in the range of the mantissa (as used by decodeFloat/encodeFloat). --- Then round the rational to an Integer and encode it with the exponent --- that we got from the scaling. --- To speed up the scaling process we compute the log2 of the number to get --- a first guess of the exponent. - -fromRat' :: (RealFloat a) => Rational -> a -fromRat' x = r - where b = floatRadix r - p = floatDigits r - (minExp0, _) = floatRange r - minExp = minExp0 - p -- the real minimum exponent - xMin = toRational (expt b (p-1)) - xMax = toRational (expt b p) - p0 = (integerLogBase b (numerator x) - integerLogBase b (denominator x) - p) `max` minExp - f = if p0 < 0 then 1 % expt b (-p0) else expt b p0 % 1 - (x', p') = scaleRat (toRational b) minExp xMin xMax p0 (x / f) - r = encodeFloat (round x') p' - --- Scale x until xMin <= x < xMax, or p (the exponent) <= minExp. -scaleRat :: Rational -> Int -> Rational -> Rational -> Int -> Rational -> (Rational, Int) -scaleRat b minExp xMin xMax p x = - if p <= minExp then - (x, p) - else if x >= xMax then - scaleRat b minExp xMin xMax (p+1) (x/b) - else if x < xMin then - scaleRat b minExp xMin xMax (p-1) (x*b) - else - (x, p) - --- Exponentiation with a cache for the most common numbers. -minExpt = 0::Int -maxExpt = 1100::Int -expt :: Integer -> Int -> Integer -expt base n = - if base == 2 && n >= minExpt && n <= maxExpt then - expts!n - else - base^n -expts :: Array Int Integer -expts = array (minExpt,maxExpt) [(n,2^n) | n <- [minExpt .. maxExpt]] - --- Compute the (floor of the) log of i in base b. --- Simplest way would be just divide i by b until it's smaller then b, but that would --- be very slow! We are just slightly more clever. -integerLogBase :: Integer -> Integer -> Int -integerLogBase b i = - if i < b then - 0 - else - -- Try squaring the base first to cut down the number of divisions. - let l = 2 * integerLogBase (b*b) i - doDiv :: Integer -> Int -> Int - doDiv i l = if i < b then l else doDiv (i `div` b) (l+1) - in doDiv (i `div` (b^l)) l -\end{code} - -%********************************************************* -%* * -\subsection{Numeric primops} -%* * -%********************************************************* - -Definitions of the boxed PrimOps; these will be -used in the case of partial applications, etc. - -\begin{code} -plusFloat (F# x) (F# y) = F# (plusFloat# x y) -minusFloat (F# x) (F# y) = F# (minusFloat# x y) -timesFloat (F# x) (F# y) = F# (timesFloat# x y) -divideFloat (F# x) (F# y) = F# (divideFloat# x y) -negateFloat (F# x) = F# (negateFloat# x) - -gtFloat (F# x) (F# y) = gtFloat# x y -geFloat (F# x) (F# y) = geFloat# x y -eqFloat (F# x) (F# y) = eqFloat# x y -neFloat (F# x) (F# y) = neFloat# x y -ltFloat (F# x) (F# y) = ltFloat# x y -leFloat (F# x) (F# y) = leFloat# x y - -float2Int (F# x) = I# (float2Int# x) -int2Float (I# x) = F# (int2Float# x) - -expFloat (F# x) = F# (expFloat# x) -logFloat (F# x) = F# (logFloat# x) -sqrtFloat (F# x) = F# (sqrtFloat# x) -sinFloat (F# x) = F# (sinFloat# x) -cosFloat (F# x) = F# (cosFloat# x) -tanFloat (F# x) = F# (tanFloat# x) -asinFloat (F# x) = F# (asinFloat# x) -acosFloat (F# x) = F# (acosFloat# x) -atanFloat (F# x) = F# (atanFloat# x) -sinhFloat (F# x) = F# (sinhFloat# x) -coshFloat (F# x) = F# (coshFloat# x) -tanhFloat (F# x) = F# (tanhFloat# x) - -powerFloat (F# x) (F# y) = F# (powerFloat# x y) - --- definitions of the boxed PrimOps; these will be --- used in the case of partial applications, etc. - -plusDouble (D# x) (D# y) = D# (x +## y) -minusDouble (D# x) (D# y) = D# (x -## y) -timesDouble (D# x) (D# y) = D# (x *## y) -divideDouble (D# x) (D# y) = D# (x /## y) -negateDouble (D# x) = D# (negateDouble# x) - -gtDouble (D# x) (D# y) = x >## y -geDouble (D# x) (D# y) = x >=## y -eqDouble (D# x) (D# y) = x ==## y -neDouble (D# x) (D# y) = x /=## y -ltDouble (D# x) (D# y) = x <## y -leDouble (D# x) (D# y) = x <=## y - -double2Int (D# x) = I# (double2Int# x) -int2Double (I# x) = D# (int2Double# x) -double2Float (D# x) = F# (double2Float# x) -float2Double (F# x) = D# (float2Double# x) - -expDouble (D# x) = D# (expDouble# x) -logDouble (D# x) = D# (logDouble# x) -sqrtDouble (D# x) = D# (sqrtDouble# x) -sinDouble (D# x) = D# (sinDouble# x) -cosDouble (D# x) = D# (cosDouble# x) -tanDouble (D# x) = D# (tanDouble# x) -asinDouble (D# x) = D# (asinDouble# x) -acosDouble (D# x) = D# (acosDouble# x) -atanDouble (D# x) = D# (atanDouble# x) -sinhDouble (D# x) = D# (sinhDouble# x) -coshDouble (D# x) = D# (coshDouble# x) -tanhDouble (D# x) = D# (tanhDouble# x) - -powerDouble (D# x) (D# y) = D# (x **## y) -\end{code}