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 PrelBase
+import GHC
import {-# SOURCE #-} IOBase ( error )
import PrelList
-import PrelBase
+
import ArrBase ( Array, array, (!) )
+import UnsafeST ( unsafePerformPrimIO )
import Ix ( Ix(..) )
-import GHC
+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
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
%* *
%*********************************************************
-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
+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
+ 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
+ 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}
their greatest common divisor.
\begin{code}
-reduce _ 0 = error "{Ratio.%}: zero denominator"
+reduce x 0 = error "{Ratio.%}: zero denominator"
reduce x y = (x `quot` d) :% (y `quot` d)
where d = gcd x y
\end{code}
| 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}
+\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
\begin{pseudocode}
{-# GENERATE_SPECS fromRational__ a{Double#,Double} #-}
-fromRational__ :: (RealFloat a) => Rational -> a
-fromRational__ x = x'
+fromRat :: (RealFloat a) => Rational -> a
+fromRat x = x'
where x' = f e
-- If the exponent of the nearest floating-point number to x
Now, here's Lennart's code.
\begin{code}
-fromRational__ :: (RealFloat a) => Rational -> a
-fromRational__ x =
+--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
in doDiv (i `div` (b^l)) l
\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'
- }}
-
-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)
-\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.
-
-\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
-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}
%*********************************************************
%* *
projects / ghc-hetmet.git / blobdiff