+% -----------------------------------------------------------------------------
+% $Id: Numeric.lhs,v 1.13 2001/02/28 00:01:03 qrczak Exp $
%
-% (c) The AQUA Project, Glasgow University, 1997-98
+% (c) The University of Glasgow, 1997-2000
%
+
\section[Numeric]{Numeric interface}
Odds and ends, mostly functions for reading and showing
\begin{code}
-{-# OPTIONS -fno-implicit-prelude #-}
module Numeric
- (
- fromRat,
- showSigned,
- readSigned,
- showInt,
- readInt,
-
- readDec, readOct, readHex,
-
- showEFloat,
- showFFloat,
- showGFloat,
- showFloat,
- readFloat,
-
- floatToDigits,
- lexDigits
-
- ) where
-
-import PrelBase
-import PrelMaybe
-import PrelArr
-import PrelNum
-import PrelRead
-
-\end{code}
-
-%*********************************************************
-%* *
-\subsection[Numeric-signatures]{Signatures}
-%* *
-%*********************************************************
-Interface on offer:
+ ( fromRat -- :: (RealFloat a) => Rational -> a
+ , showSigned -- :: (Real a) => (a -> ShowS) -> Int -> a -> ShowS
+ , readSigned -- :: (Real a) => ReadS a -> ReadS a
+ , showInt -- :: Integral a => a -> ShowS
+ , readInt -- :: (Integral a) => a -> (Char -> Bool) -> (Char -> Int) -> ReadS a
+
+ , readDec -- :: (Integral a) => ReadS a
+ , readOct -- :: (Integral a) => ReadS a
+ , readHex -- :: (Integral a) => ReadS a
-\begin{pseudocode}
-fromRat :: (RealFloat a) => Rational -> a
+ , showEFloat -- :: (RealFloat a) => Maybe Int -> a -> ShowS
+ , showFFloat -- :: (RealFloat a) => Maybe Int -> a -> ShowS
+ , showGFloat -- :: (RealFloat a) => Maybe Int -> a -> ShowS
+ , showFloat -- :: (RealFloat a) => a -> ShowS
+ , readFloat -- :: (RealFloat a) => ReadS a
+
+
+ , floatToDigits -- :: (RealFloat a) => Integer -> a -> ([Int], Int)
+ , lexDigits -- :: ReadS String
-showSigned :: (Real a) => (a -> ShowS) -> Int -> a -> ShowS
-readSigned :: (Real a) => ReadS a -> ReadS a
+ -- Implementation checked wrt. Haskell 98 lib report, 1/99.
+ ) where
-showInt :: Integral a => a -> ShowS
-readInt :: (Integral a) => a -> (Char -> Bool) -> (Char -> Int) -> ReadS a
+import Char
-readDec :: (Integral a) => ReadS a
-readOct :: (Integral a) => ReadS a
-readHex :: (Integral a) => ReadS a
+#ifndef __HUGS__
+ -- GHC imports
+import Prelude -- For dependencies
+import PrelBase ( Char(..), unsafeChr )
+import PrelRead -- Lots of things
+import PrelReal ( showSigned )
+import PrelFloat ( fromRat, FFFormat(..),
+ formatRealFloat, floatToDigits, showFloat
+ )
+#else
+ -- Hugs imports
+import Array
+#endif
-showEFloat :: (RealFloat a) => Maybe Int -> a -> ShowS
-showFFloat :: (RealFloat a) => Maybe Int -> a -> ShowS
-showGFloat :: (RealFloat a) => Maybe Int -> a -> ShowS
-showFloat :: (RealFloat a) => a -> ShowS
-
-readFloat :: (RealFloat a) => ReadS a
+\end{code}
-floatToDigits :: (RealFloat a) => Integer -> a -> ([Int], Int)
-lexDigits :: ReadS String
-\end{pseudocode}
+#ifndef __HUGS__
\begin{code}
showInt :: Integral a => a -> ShowS
-showInt 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 showInt n' r'
- }}
+showInt n cs
+ | n < 0 = error "Numeric.showInt: can't show negative numbers"
+ | otherwise = go n cs
+ where
+ go n cs
+ | n < 10 = case unsafeChr (ord '0' + fromIntegral n) of
+ c@(C# _) -> c:cs
+ | otherwise = case unsafeChr (ord '0' + fromIntegral r) of
+ c@(C# _) -> go q (c:cs)
+ where
+ (q,r) = n `quotRem` 10
\end{code}
Controlling the format and precision of floats. The code that
mutual module deps.
\begin{code}
+{-# SPECIALIZE showEFloat ::
+ Maybe Int -> Float -> ShowS,
+ Maybe Int -> Double -> ShowS #-}
+{-# SPECIALIZE showFFloat ::
+ Maybe Int -> Float -> ShowS,
+ Maybe Int -> Double -> ShowS #-}
+{-# SPECIALIZE showGFloat ::
+ Maybe Int -> Float -> ShowS,
+ Maybe Int -> Double -> ShowS #-}
+
showEFloat :: (RealFloat a) => Maybe Int -> a -> ShowS
showFFloat :: (RealFloat a) => Maybe Int -> a -> ShowS
showGFloat :: (RealFloat a) => Maybe Int -> a -> ShowS
showGFloat d x = showString (formatRealFloat FFGeneric d x)
\end{code}
+
+#else
+
+%*********************************************************
+%* *
+ All of this code is for Hugs only
+ GHC gets it from PrelFloat!
+%* *
+%*********************************************************
+
+\begin{code}
+-- This converts a rational to a floating. This should be used in the
+-- Fractional instances of Float and Double.
+
+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
+
+
+-- Misc utilities to show integers and floats
+
+showEFloat :: (RealFloat a) => Maybe Int -> a -> ShowS
+showFFloat :: (RealFloat a) => Maybe Int -> a -> ShowS
+showGFloat :: (RealFloat a) => Maybe Int -> a -> ShowS
+showFloat :: (RealFloat a) => a -> ShowS
+
+showEFloat d x = showString (formatRealFloat FFExponent d x)
+showFFloat d x = showString (formatRealFloat FFFixed d x)
+showGFloat d x = showString (formatRealFloat FFGeneric d x)
+showFloat = showGFloat Nothing
+
+-- These are the format types. This type is not exported.
+
+data FFFormat = FFExponent | FFFixed | FFGeneric
+
+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 ->
+ case ds of
+ ['0'] -> "0.0e0"
+ [d] -> d : ".0e" ++ show (e-1)
+ d:ds -> d : '.' : ds ++ 'e':show (e-1)
+ 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 ->
+ case decs of
+ Nothing ->
+ let f 0 s ds = mk0 s ++ "." ++ mk0 ds
+ f n s "" = f (n-1) (s++"0") ""
+ f n s (d:ds) = f (n-1) (s++[d]) ds
+ mk0 "" = "0"
+ mk0 s = s
+ in f e "" ds
+ Just dec ->
+ let dec' = max dec 0 in
+ if e >= 0 then
+ let (ei, is') = roundTo base (dec' + e) is
+ (ls, rs) = splitAt (e+ei) (map intToDigit is')
+ in (if null ls then "0" else 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 = case f d is of
+ (0, is) -> (0, is)
+ (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)) +
+ fromIntegral 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 fromIntegral (reverse rds), k)
+\end{code}
+#endif
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