[project @ 2002-06-05 14:08:24 by simonpj]
[packages/base.git] / Numeric.hs
1 {-# OPTIONS -fno-implicit-prelude #-}
2 -----------------------------------------------------------------------------
3 -- |
4 -- Module : Numeric
5 -- Copyright : (c) The University of Glasgow 2002
6 -- License : BSD-style (see the file libraries/base/LICENSE)
7 --
8 -- Maintainer : libraries@haskell.org
9 -- Stability : provisional
10 -- Portability : portable
11 --
12 -- Odds and ends, mostly functions for reading and showing
13 -- RealFloat-like kind of values.
14 --
15 -----------------------------------------------------------------------------
16
17 module Numeric (
18
19 fromRat, -- :: (RealFloat a) => Rational -> a
20 showSigned, -- :: (Real a) => (a -> ShowS) -> Int -> a -> ShowS
21 readSigned, -- :: (Real a) => ReadS a -> ReadS a
22
23 readInt, -- :: (Integral a) => a -> (Char -> Bool)
24 -- -> (Char -> Int) -> ReadS a
25 readDec, -- :: (Integral a) => ReadS a
26 readOct, -- :: (Integral a) => ReadS a
27 readHex, -- :: (Integral a) => ReadS a
28
29 showInt, -- :: Integral a => a -> ShowS
30 showIntAtBase, -- :: Integral a => a -> (a -> Char) -> a -> ShowS
31 showHex, -- :: Integral a => a -> ShowS
32 showOct, -- :: Integral a => a -> ShowS
33 showBin, -- :: Integral a => a -> ShowS
34
35 showEFloat, -- :: (RealFloat a) => Maybe Int -> a -> ShowS
36 showFFloat, -- :: (RealFloat a) => Maybe Int -> a -> ShowS
37 showGFloat, -- :: (RealFloat a) => Maybe Int -> a -> ShowS
38 showFloat, -- :: (RealFloat a) => a -> ShowS
39 readFloat, -- :: (RealFloat a) => ReadS a
40
41 floatToDigits, -- :: (RealFloat a) => Integer -> a -> ([Int], Int)
42 lexDigits, -- :: ReadS String
43
44 ) where
45
46 import Data.Char
47
48 #ifdef __GLASGOW_HASKELL__
49 import GHC.Base
50 import GHC.Read
51 import GHC.Real
52 import GHC.Float
53 import GHC.Num
54 import GHC.Show
55 import Data.Maybe
56 import Text.ParserCombinators.ReadP( ReadP, readP_to_S, pfail )
57 import qualified Text.Read.Lex as L
58 #endif
59
60 #ifdef __HUGS__
61 import Array
62 #endif
63
64
65 -- -----------------------------------------------------------------------------
66 -- Reading
67
68 readInt :: Num a => a -> (Char -> Bool) -> (Char -> Int) -> ReadS a
69 readInt base isDigit valDigit = readP_to_S (L.readIntP base isDigit valDigit)
70
71 readOct, readDec, readHex :: Num a => ReadS a
72 readOct = readP_to_S L.readOctP
73 readDec = readP_to_S L.readDecP
74 readHex = readP_to_S L.readHexP
75
76 readFloat :: RealFrac a => ReadS a
77 readFloat = readP_to_S readFloatP
78
79 readFloatP :: RealFrac a => ReadP a
80 readFloatP =
81 do tok <- L.lex
82 case tok of
83 L.Rat y -> return (fromRational y)
84 L.Int i -> return (fromInteger i)
85 other -> pfail
86
87 -- It's turgid to have readSigned work using list comprehensions,
88 -- but it's specified as a ReadS to ReadS transformer
89 -- With a bit of luck no one will use it.
90 readSigned :: (Real a) => ReadS a -> ReadS a
91 readSigned readPos = readParen False read'
92 where read' r = read'' r ++
93 (do
94 ("-",s) <- lex r
95 (x,t) <- read'' s
96 return (-x,t))
97 read'' r = do
98 (str,s) <- lex r
99 (n,"") <- readPos str
100 return (n,s)
101
102
103 -- -----------------------------------------------------------------------------
104 -- Showing
105
106 #ifdef __GLASGOW_HASKELL__
107 showInt :: Integral a => a -> ShowS
108 showInt n cs
109 | n < 0 = error "Numeric.showInt: can't show negative numbers"
110 | otherwise = go n cs
111 where
112 go n cs
113 | n < 10 = case unsafeChr (ord '0' + fromIntegral n) of
114 c@(C# _) -> c:cs
115 | otherwise = case unsafeChr (ord '0' + fromIntegral r) of
116 c@(C# _) -> go q (c:cs)
117 where
118 (q,r) = n `quotRem` 10
119
120 -- Controlling the format and precision of floats. The code that
121 -- implements the formatting itself is in @PrelNum@ to avoid
122 -- mutual module deps.
123
124 {-# SPECIALIZE showEFloat ::
125 Maybe Int -> Float -> ShowS,
126 Maybe Int -> Double -> ShowS #-}
127 {-# SPECIALIZE showFFloat ::
128 Maybe Int -> Float -> ShowS,
129 Maybe Int -> Double -> ShowS #-}
130 {-# SPECIALIZE showGFloat ::
131 Maybe Int -> Float -> ShowS,
132 Maybe Int -> Double -> ShowS #-}
133
134 showEFloat :: (RealFloat a) => Maybe Int -> a -> ShowS
135 showFFloat :: (RealFloat a) => Maybe Int -> a -> ShowS
136 showGFloat :: (RealFloat a) => Maybe Int -> a -> ShowS
137
138 showEFloat d x = showString (formatRealFloat FFExponent d x)
139 showFFloat d x = showString (formatRealFloat FFFixed d x)
140 showGFloat d x = showString (formatRealFloat FFGeneric d x)
141 #endif
142
143 #ifdef __HUGS__
144 -- This converts a rational to a floating. This should be used in the
145 -- Fractional instances of Float and Double.
146
147 fromRat :: (RealFloat a) => Rational -> a
148 fromRat x =
149 if x == 0 then encodeFloat 0 0 -- Handle exceptional cases
150 else if x < 0 then - fromRat' (-x) -- first.
151 else fromRat' x
152
153 -- Conversion process:
154 -- Scale the rational number by the RealFloat base until
155 -- it lies in the range of the mantissa (as used by decodeFloat/encodeFloat).
156 -- Then round the rational to an Integer and encode it with the exponent
157 -- that we got from the scaling.
158 -- To speed up the scaling process we compute the log2 of the number to get
159 -- a first guess of the exponent.
160 fromRat' :: (RealFloat a) => Rational -> a
161 fromRat' x = r
162 where b = floatRadix r
163 p = floatDigits r
164 (minExp0, _) = floatRange r
165 minExp = minExp0 - p -- the real minimum exponent
166 xMin = toRational (expt b (p-1))
167 xMax = toRational (expt b p)
168 p0 = (integerLogBase b (numerator x) -
169 integerLogBase b (denominator x) - p) `max` minExp
170 f = if p0 < 0 then 1 % expt b (-p0) else expt b p0 % 1
171 (x', p') = scaleRat (toRational b) minExp xMin xMax p0 (x / f)
172 r = encodeFloat (round x') p'
173
174 -- Scale x until xMin <= x < xMax, or p (the exponent) <= minExp.
175 scaleRat :: Rational -> Int -> Rational -> Rational ->
176 Int -> Rational -> (Rational, Int)
177 scaleRat b minExp xMin xMax p x =
178 if p <= minExp then
179 (x, p)
180 else if x >= xMax then
181 scaleRat b minExp xMin xMax (p+1) (x/b)
182 else if x < xMin then
183 scaleRat b minExp xMin xMax (p-1) (x*b)
184 else
185 (x, p)
186
187 -- Exponentiation with a cache for the most common numbers.
188 minExpt = 0::Int
189 maxExpt = 1100::Int
190 expt :: Integer -> Int -> Integer
191 expt base n =
192 if base == 2 && n >= minExpt && n <= maxExpt then
193 expts!n
194 else
195 base^n
196
197 expts :: Array Int Integer
198 expts = array (minExpt,maxExpt) [(n,2^n) | n <- [minExpt .. maxExpt]]
199
200 -- Compute the (floor of the) log of i in base b.
201 -- Simplest way would be just divide i by b until it's smaller then b,
202 -- but that would be very slow! We are just slightly more clever.
203 integerLogBase :: Integer -> Integer -> Int
204 integerLogBase b i =
205 if i < b then
206 0
207 else
208 -- Try squaring the base first to cut down the number of divisions.
209 let l = 2 * integerLogBase (b*b) i
210 doDiv :: Integer -> Int -> Int
211 doDiv i l = if i < b then l else doDiv (i `div` b) (l+1)
212 in doDiv (i `div` (b^l)) l
213
214
215 -- Misc utilities to show integers and floats
216
217 showEFloat :: (RealFloat a) => Maybe Int -> a -> ShowS
218 showFFloat :: (RealFloat a) => Maybe Int -> a -> ShowS
219 showGFloat :: (RealFloat a) => Maybe Int -> a -> ShowS
220 showFloat :: (RealFloat a) => a -> ShowS
221
222 showEFloat d x = showString (formatRealFloat FFExponent d x)
223 showFFloat d x = showString (formatRealFloat FFFixed d x)
224 showGFloat d x = showString (formatRealFloat FFGeneric d x)
225 showFloat = showGFloat Nothing
226
227 -- These are the format types. This type is not exported.
228
229 data FFFormat = FFExponent | FFFixed | FFGeneric
230
231 formatRealFloat :: (RealFloat a) => FFFormat -> Maybe Int -> a -> String
232 formatRealFloat fmt decs x = s
233 where base = 10
234 s = if isNaN x then
235 "NaN"
236 else if isInfinite x then
237 if x < 0 then "-Infinity" else "Infinity"
238 else if x < 0 || isNegativeZero x then
239 '-' : doFmt fmt (floatToDigits (toInteger base) (-x))
240 else
241 doFmt fmt (floatToDigits (toInteger base) x)
242 doFmt fmt (is, e) =
243 let ds = map intToDigit is
244 in case fmt of
245 FFGeneric ->
246 doFmt (if e < 0 || e > 7 then FFExponent else FFFixed)
247 (is, e)
248 FFExponent ->
249 case decs of
250 Nothing ->
251 case ds of
252 ['0'] -> "0.0e0"
253 [d] -> d : ".0e" ++ show (e-1)
254 d:ds -> d : '.' : ds ++ 'e':show (e-1)
255 Just dec ->
256 let dec' = max dec 1 in
257 case is of
258 [0] -> '0':'.':take dec' (repeat '0') ++ "e0"
259 _ ->
260 let (ei, is') = roundTo base (dec'+1) is
261 d:ds = map intToDigit
262 (if ei > 0 then init is' else is')
263 in d:'.':ds ++ "e" ++ show (e-1+ei)
264 FFFixed ->
265 case decs of
266 Nothing ->
267 let f 0 s ds = mk0 s ++ "." ++ mk0 ds
268 f n s "" = f (n-1) (s++"0") ""
269 f n s (d:ds) = f (n-1) (s++[d]) ds
270 mk0 "" = "0"
271 mk0 s = s
272 in f e "" ds
273 Just dec ->
274 let dec' = max dec 0 in
275 if e >= 0 then
276 let (ei, is') = roundTo base (dec' + e) is
277 (ls, rs) = splitAt (e+ei) (map intToDigit is')
278 in (if null ls then "0" else ls) ++
279 (if null rs then "" else '.' : rs)
280 else
281 let (ei, is') = roundTo base dec'
282 (replicate (-e) 0 ++ is)
283 d : ds = map intToDigit
284 (if ei > 0 then is' else 0:is')
285 in d : '.' : ds
286
287 roundTo :: Int -> Int -> [Int] -> (Int, [Int])
288 roundTo base d is = case f d is of
289 (0, is) -> (0, is)
290 (1, is) -> (1, 1 : is)
291 where b2 = base `div` 2
292 f n [] = (0, replicate n 0)
293 f 0 (i:_) = (if i >= b2 then 1 else 0, [])
294 f d (i:is) =
295 let (c, ds) = f (d-1) is
296 i' = c + i
297 in if i' == base then (1, 0:ds) else (0, i':ds)
298
299 --
300 -- Based on "Printing Floating-Point Numbers Quickly and Accurately"
301 -- by R.G. Burger and R. K. Dybvig, in PLDI 96.
302 -- This version uses a much slower logarithm estimator. It should be improved.
303
304 -- This function returns a list of digits (Ints in [0..base-1]) and an
305 -- exponent.
306
307 floatToDigits :: (RealFloat a) => Integer -> a -> ([Int], Int)
308
309 floatToDigits _ 0 = ([0], 0)
310 floatToDigits base x =
311 let (f0, e0) = decodeFloat x
312 (minExp0, _) = floatRange x
313 p = floatDigits x
314 b = floatRadix x
315 minExp = minExp0 - p -- the real minimum exponent
316 -- Haskell requires that f be adjusted so denormalized numbers
317 -- will have an impossibly low exponent. Adjust for this.
318 (f, e) = let n = minExp - e0
319 in if n > 0 then (f0 `div` (b^n), e0+n) else (f0, e0)
320
321 (r, s, mUp, mDn) =
322 if e >= 0 then
323 let be = b^e in
324 if f == b^(p-1) then
325 (f*be*b*2, 2*b, be*b, b)
326 else
327 (f*be*2, 2, be, be)
328 else
329 if e > minExp && f == b^(p-1) then
330 (f*b*2, b^(-e+1)*2, b, 1)
331 else
332 (f*2, b^(-e)*2, 1, 1)
333 k =
334 let k0 =
335 if b==2 && base==10 then
336 -- logBase 10 2 is slightly bigger than 3/10 so
337 -- the following will err on the low side. Ignoring
338 -- the fraction will make it err even more.
339 -- Haskell promises that p-1 <= logBase b f < p.
340 (p - 1 + e0) * 3 `div` 10
341 else
342 ceiling ((log (fromInteger (f+1)) +
343 fromIntegral e * log (fromInteger b)) /
344 log (fromInteger base))
345 fixup n =
346 if n >= 0 then
347 if r + mUp <= expt base n * s then n else fixup (n+1)
348 else
349 if expt base (-n) * (r + mUp) <= s then n
350 else fixup (n+1)
351 in fixup k0
352
353 gen ds rn sN mUpN mDnN =
354 let (dn, rn') = (rn * base) `divMod` sN
355 mUpN' = mUpN * base
356 mDnN' = mDnN * base
357 in case (rn' < mDnN', rn' + mUpN' > sN) of
358 (True, False) -> dn : ds
359 (False, True) -> dn+1 : ds
360 (True, True) -> if rn' * 2 < sN then dn : ds else dn+1 : ds
361 (False, False) -> gen (dn:ds) rn' sN mUpN' mDnN'
362 rds =
363 if k >= 0 then
364 gen [] r (s * expt base k) mUp mDn
365 else
366 let bk = expt base (-k)
367 in gen [] (r * bk) s (mUp * bk) (mDn * bk)
368 in (map fromIntegral (reverse rds), k)
369 #endif
370
371 -- ---------------------------------------------------------------------------
372 -- Integer printing functions
373
374 showIntAtBase :: Integral a => a -> (a -> Char) -> a -> ShowS
375 showIntAtBase base toChr n r
376 | n < 0 = error ("Numeric.showIntAtBase: applied to negative number " ++ show n)
377 | otherwise =
378 case quotRem n base of { (n', d) ->
379 let c = toChr d in
380 seq c $ -- stricter than necessary
381 let
382 r' = c : r
383 in
384 if n' == 0 then r' else showIntAtBase base toChr n' r'
385 }
386
387 showHex :: Integral a => a -> ShowS
388 showHex n r =
389 showString "0x" $
390 showIntAtBase 16 (toChrHex) n r
391 where
392 toChrHex d
393 | d < 10 = chr (ord '0' + fromIntegral d)
394 | otherwise = chr (ord 'a' + fromIntegral (d - 10))
395
396 showOct :: Integral a => a -> ShowS
397 showOct n r =
398 showString "0o" $
399 showIntAtBase 8 (toChrOct) n r
400 where toChrOct d = chr (ord '0' + fromIntegral d)
401
402 showBin :: Integral a => a -> ShowS
403 showBin n r =
404 showString "0b" $
405 showIntAtBase 2 (toChrOct) n r
406 where toChrOct d = chr (ord '0' + fromIntegral d)