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}
-{-# OPTIONS -fno-implicit-prelude #-}
-
module PrelNum where
-import {-# SOURCE #-} IOBase ( error )
-import PrelList
import PrelBase
import GHC
+import {-# SOURCE #-} IOBase ( error )
+import PrelList
+
+import ArrBase ( Array, array, (!) )
+import STBase ( unsafePerformPrimIO )
+import Ix ( Ix(..) )
+import Foreign () -- This import tells the dependency analyser to compile Foreign first.
+ -- There's an implicit dependency on Foreign because the ccalls in
+ -- PrelNum implicitly mention CCallable.
infixr 8 ^, ^^, **
infixl 7 /, %, `quot`, `rem`, `div`, `mod`
-- 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
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 = 0; integer_1 = 1; integer_2 = 2; integer_m1 = -1
+integer_0 = int2Integer# 0#
+integer_1 = int2Integer# 1#
+integer_2 = int2Integer# 2#
+integer_m1 = int2Integer# (negateInt# 1#)
\end{code}
(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 `geFloat#` y
+ (F# x) > (F# y) = x `gtFloat#` y
instance Num Float where
(+) x y = plusFloat x y
instance Fractional Float where
(/) x y = divideFloat x y
- fromRational x = fromRational__ x
+ fromRational x = fromRat x
recip x = 1.0 / x
instance Floating Float where
scaleFloat k x = case decodeFloat x of
(m,n) -> encodeFloat m (n+k)
+ isNaN x =
+ (0::Int) /= unsafePerformPrimIO (_ccall_ isFloatNaN x) {- a _pure_function! -}
+ isInfinite x =
+ (0::Int) /= unsafePerformPrimIO (_ccall_ isFloatInfinite x) {- ditto! -}
+ isDenormalized x =
+ (0::Int) /= unsafePerformPrimIO (_ccall_ isFloatDenormalized x) -- ..
+ isNegativeZero x =
+ (0::Int) /= unsafePerformPrimIO (_ccall_ isFloatNegativeZero x) -- ...
+ isIEEE x = True
instance Show Float where
showsPrec x = showSigned showFloat x
instance Fractional Double where
(/) x y = divideDouble x y
- fromRational x = fromRational__ x
+ fromRational x = fromRat x
recip x = 1.0 / x
instance Floating Double where
scaleFloat k x = case decodeFloat x of
(m,n) -> encodeFloat m (n+k)
+ isNaN x =
+ (0::Int) /= unsafePerformPrimIO (_ccall_ isDoubleNaN x) {- a _pure_function! -}
+ isInfinite x =
+ (0::Int) /= unsafePerformPrimIO (_ccall_ isDoubleInfinite x) {- ditto -}
+ isDenormalized x =
+ (0::Int) /= unsafePerformPrimIO (_ccall_ isDoubleDenormalized x) -- ..
+ isNegativeZero x =
+ (0::Int) /= unsafePerformPrimIO (_ccall_ isDoubleNegativeZero x) -- ...
+ isIEEE x = True
instance Show Double where
showsPrec x = showSigned showFloat x
%*********************************************************
\begin{code}
-data (Integral a) => Ratio a = a :% a deriving (Eq)
+data (Integral a) => Ratio a = !a :% !a deriving (Eq)
type Rational = Ratio Integer
\end{code}
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.
-reduce _ 0 = error "{Ratio.%}: zero denominator"
+\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
| x > 0 = simplest' n d n' d'
| y < 0 = - simplest' (-n') d' (-n) d
| otherwise = 0 :% 1
- where xr@(n:%d) = toRational x
- (n':%d') = toRational y
+ 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
| otherwise = (q*n''+d'') :% n''
where (q,r) = quotRem n d
(q',r') = quotRem n' d'
- (n'':%d'') = simplest' d' r' d r
+ nd'' = simplest' d' r' d r
+ n'' = numerator nd''
+ d'' = denominator nd''
\end{code}
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
(shows x . showString " % " . shows y)
\end{code}
-{-
-[In response to a request by simonpj, 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.
-
-How's this?
-
-Joe
--}
-
\begin{code}
-{-# GENERATE_SPECS fromRational__ a{Double#,Double} #-}
-fromRational__ :: (RealFloat a) => Rational -> a
-fromRational__ 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{code}
-
-
-%*********************************************************
-%* *
-\subsection{Showing numbers}
-%* *
-%*********************************************************
-
-\begin{code}
-showInteger n r
- = case quotRem n 10 of { (n', d) ->
- case (chr (ord_0 + fromIntegral d)) of { C# c# -> -- stricter than necessary
- let
- r' = C# c# : r
- in
- if n' == 0 then r' else showInteger n' r'
- }}
-
+--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))
chr (fromInteger (n + ord_0)) : cs
else
jtos' (n `quot` 10) (chr (fromInteger (n `rem` 10 + ord_0)) : cs)
-\end{code}
-The functions showFloat below uses rational arithmetic
-to insure correct conversion between the floating-point radix and
-decimal. It is often possible to use a higher-precision floating-
-point type to obtain the same results.
+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)
-\begin{code}
-{-# GENERATE_SPECS showFloat a{Double#,Double} #-}
-showFloat:: (RealFloat a) => a -> ShowS
-showFloat x =
- if x == 0 then showString ("0." ++ take (m-1) zeros)
- else if e >= m-1 || e < 0 then showSci else showFix
- where
- showFix = showString whole . showChar '.' . showString frac
- where (whole,frac) = splitAt (e+1) (show sig)
- showSci = showChar d . showChar '.' . showString frac
- . showChar 'e' . shows e
- where (d:frac) = show sig
- (m, sig, e) = if b == 10 then (w, s, n+w-1)
- else (m', sig', e' )
- m' = ceiling
- ((fromInt w * log (fromInteger b)) / log 10 :: Double)
- + 1
- (sig', e') = if sig1 >= 10^m' then (round (t/10), e1+1)
- else if sig1 < 10^(m'-1) then (round (t*10), e1-1)
- else (sig1, e1 )
- sig1 = round t
- t = s%1 * (b%1)^^n * 10^^(m'-e1-1)
- e1 = floor (logBase 10 x)
- (s, n) = decodeFloat x
- b = floatRadix x
- w = floatDigits x
-
-zeros = repeat '0'
\end{code}
@showRational@ converts a Rational to a string that looks like a
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}
projects / ghc-hetmet.git / blobdiff