\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 #-}
+{-# OPTIONS -fno-implicit-prelude #-}
-#include "../includes/ieee-flpt.h"
-
-\end{code}
-
-\begin{code}
module PrelNum where
import PrelBase
-import PrelGHC
-import {-# SOURCE #-} PrelErr ( error )
-import PrelList
-import PrelMaybe
-
-import PrelArr ( Array, array, (!) )
-import PrelIOBase ( unsafePerformIO )
-import Ix ( Ix(..) )
-import PrelCCall () -- we need the definitions of CCallable and
- -- CReturnable for the _ccall_s herein.
-
+import Ix
+import {-# SOURCE #-} PrelErr
infixr 8 ^, ^^, **
-infixl 7 /, %, `quot`, `rem`, `div`, `mod`
+infixl 7 %, /, `quot`, `rem`, `div`, `mod`
\end{code}
-
%*********************************************************
%* *
\subsection{Standard numeric classes}
%*********************************************************
%* *
-\subsection{Overloaded numeric functions}
-%* *
-%*********************************************************
-
-\begin{code}
-even, odd :: (Integral a) => a -> Bool
-even n = n `rem` 2 == 0
-odd = not . even
-
-{-# SPECIALISE gcd ::
- Int -> Int -> Int,
- Integer -> Integer -> 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)
-
-{-# SPECIALISE lcm ::
- Int -> Int -> Int,
- Integer -> Integer -> Integer #-}
-lcm :: (Integral a) => a -> a -> a
-lcm _ 0 = 0
-lcm 0 _ = 0
-lcm x y = abs ((x `quot` (gcd x y)) * y)
-
-{-# SPECIALISE (^) ::
- Integer -> Integer -> Integer,
- Integer -> Int -> Integer,
- Int -> Int -> Int #-}
-(^) :: (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"
-
-{-# SPECIALISE (^^) ::
- Double -> Int -> Double,
- Rational -> Int -> Rational #-}
-(^^) :: (Fractional a, Integral b) => a -> b -> a
-x ^^ n = if n >= 0 then x^n else recip (x^(negate n))
-
-{-# SPECIALIZE fromIntegral ::
- Int -> Rational,
- Integer -> Rational,
- Int -> Int,
- Int -> Integer,
- Int -> Float,
- Int -> Double,
- Integer -> Int,
- Integer -> Integer,
- Integer -> Float,
- Integer -> Double #-}
-fromIntegral :: (Integral a, Num b) => a -> b
-fromIntegral = fromInteger . toInteger
-
-{-# SPECIALIZE fromRealFrac ::
- Double -> Rational,
- Rational -> Double,
- Float -> Rational,
- Rational -> Float,
- Rational -> Rational,
- Double -> Double,
- Double -> Float,
- Float -> Float,
- Float -> Double #-}
-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@}
%* *
%*********************************************************
then a `remInt` b
else error "Integral.Int.rem{PreludeCore}: divide by 0\n"
- n `div` d
- | n > 0 && d < 0 = mk_neg (quotInt (n-d-1) d)
- | n < 0 && d > 0 = mk_neg (quotInt (n-d+1) d)
- | otherwise = quotInt n d
- where
- {-
- - the result of (integral) division is
- defined as being truncated towards
- negative infinity. (see Sec 6.3.2 of
- the Haskell 1.4 report.)
-
- - in the case of Int, if either nominator or
- denominator is negative, we adjust the nominator
- to account for the above property before
- computing the quotient.
-
- - in the case of Int, the adjustment of the
- nominator runs the risk of overflowing. If
- we make the assumption that arithmetic is
- modulo word size, and adjust the final result
- to account for this.
- -}
-
- mk_neg r
- | r <= 0 = r
- | otherwise = -(r+1)
-
+ 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
divMod x@(I# _) y@(I# _) = (x `div` y, x `mod` y)
-- Stricter. Sorry if you don't like it. (WDP 94/10)
- toInteger (I# n#) = int2Integer# n# -- give back a full-blown Integer
+--OLD: even x = eqInt (x `mod` 2) 0
+--OLD: odd x = neInt (x `mod` 2) 0
+
+ toInteger (I# i) = int2Integer i -- give back a full-blown Integer
toInt x = x
\end{code}
-
%*********************************************************
%* *
-\subsection{Type @Integer@}
+\subsection{Instances for @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#
instance Num Integer where
(+) (J# a1 s1 d1) (J# a2 s2 d2)
- = plusInteger# a1 s1 d1 a2 s2 d2
+ = case plusInteger# a1 s1 d1 a2 s2 d2 of (# a, s, d #) -> J# a s d
(-) (J# a1 s1 d1) (J# a2 s2 d2)
- = minusInteger# a1 s1 d1 a2 s2 d2
+ = case minusInteger# a1 s1 d1 a2 s2 d2 of (# a, s, d #) -> J# a s d
- negate (J# a s d) = negateInteger# a s d
+ negate (J# a s d)
+ = case negateInteger# a s d of (# a, s, d #) -> J# a s d
(*) (J# a1 s1 d1) (J# a2 s2 d2)
- = timesInteger# a1 s1 d1 a2 s2 d2
+ = case timesInteger# a1 s1 d1 a2 s2 d2 of (# a, s, d #) -> J# a s d
-- ORIG: abs n = if n >= 0 then n else -n
= case 0 of { J# a2 s2 d2 ->
if (cmpInteger# a1 s1 d1 a2 s2 d2) >=# 0#
then n
- else negateInteger# a1 s1 d1
+ else case negateInteger# a1 s1 d1 of (# a, s, d #) -> J# a s d
}
signum n@(J# a1 s1 d1)
fromInteger x = x
- fromInt (I# n#) = int2Integer# n# -- gives back a full-blown Integer
+ fromInt (I# i) = int2Integer i
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
+ (# a3, s3, d3, a4, s4, d4 #)
-> (J# a3 s3 d3, J# a4 s4 d4)
{- USING THE UNDERLYING "GMP" CODE IS DUBIOUS FOR NOW:
| 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@}
-%* *
-%*********************************************************
+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
-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.
+jtos :: Integer -> String
+jtos n
+ = if n < 0 then
+ '-' : jtos' (-n) []
+ else
+ jtos' n []
-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.)
+jtos' :: Integer -> String -> String
+jtos' n cs
+ = if n < 10 then
+ chr (fromInteger (n + ord_0)) : cs
+ else
+ jtos' q (chr (toInt r + (ord_0::Int)) : cs)
+ where
+ (q,r) = n `quotRem` 10
-\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)
+data (Integral a) => Ratio a = !a :% !a deriving (Eq)
type Rational = Ratio Integer
-\end{code}
-\begin{code}
{-# SPECIALISE (%) :: Integer -> Integer -> Rational #-}
-
(%) :: (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 .
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
-
-instance (Integral a) => Show (Ratio a) where
- showsPrec p (x:%y) = showParen (p > ratio_prec)
- (shows x . showString " % " . shows y)
-
--- defn. also used by the Read (Ratio a) instance PrelRead.
-ratio_prec :: Int
-ratio_prec = 7
-
-\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
- | x < 0 = showParen (p > 6) (showChar '-' . showPos (-x))
- | otherwise = 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
- | n < 0 = '-' : jtos' (-n) []
- | otherwise = jtos' n []
-
-jtos' :: Integer -> String -> String
-jtos' n cs
- | n < 10 = chr (fromInteger (n + ord_0)) : cs
- | otherwise = jtos' q (chr (toInt r + (ord_0::Int)) : cs)
- where
- (q,r) = n `quotRem` 10
-
-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
- base_i = toInteger base
-
- s
- | isNaN x = "NaN"
- | isInfinite x = (\ str -> if x < 0 then '-':str else str) "Infinity"
- | x < 0 || isNegativeZero x = '-' : doFmt fmt (floatToDigits base_i (-x))
- | otherwise = doFmt fmt (floatToDigits base_i 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)) /
- log (fromInteger base))
-
- 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
+%*********************************************************
+%* *
+\subsection{Overloaded numeric functions}
+%* *
+%*********************************************************
\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'
+even, odd :: (Integral a) => a -> Bool
+even n = n `rem` 2 == 0
+odd = not . even
--- 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.
+{-# SPECIALISE gcd ::
+ Int -> Int -> Int,
+ Integer -> Integer -> 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)
- (s,e) = decodeFloat (fromInteger (numerator x) `asTypeOf` x'
- / fromInteger (denominator x))
-\end{pseudocode}
+{-# SPECIALISE lcm ::
+ Int -> Int -> Int,
+ Integer -> Integer -> Integer #-}
+lcm :: (Integral a) => a -> a -> a
+lcm _ 0 = 0
+lcm 0 _ = 0
+lcm x y = abs ((x `quot` (gcd x y)) * y)
-Now, here's Lennart's code.
+{-# SPECIALISE (^) ::
+ Integer -> Integer -> Integer,
+ Integer -> Int -> Integer,
+ Int -> Int -> Int #-}
+(^) :: (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"
-\begin{code}
-{-# SPECIALISE fromRat ::
- Rational -> Double,
- Rational -> Float #-}
-
---fromRat :: (RealFloat a) => Rational -> a
-fromRat x
- | x == 0 = encodeFloat 0 0 -- Handle exceptional cases
- | x < 0 = - fromRat' (-x) -- first.
- | otherwise = 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
- | p <= minExp = (x, p)
- | x >= xMax = scaleRat b minExp xMin xMax (p+1) (x/b)
- | x < xMin = scaleRat b minExp xMin xMax (p-1) (x*b)
- | otherwise = (x, p)
-
--- Exponentiation with a cache for the most common numbers.
-minExpt = 0::Int
-maxExpt = 1100::Int
-
-expt :: Integer -> Int -> Integer
-expt base n
- | base == 2 && n >= minExpt && n <= maxExpt = expts!n
- | otherwise = 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
- | i < b = 0
- | otherwise = doDiv (i `div` (b^l)) l
- where
- -- Try squaring the base first to cut down the number of divisions.
- 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)
+{- SPECIALISE (^^) ::
+ Double -> Int -> Double,
+ Rational -> Int -> Rational #-}
+(^^) :: (Fractional a, Integral b) => a -> b -> a
+x ^^ n = if n >= 0 then x^n else recip (x^(negate n))
+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{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}
projects / ghc-hetmet.git / blobdiff