ZVON > References > Haskell reference
 Intro / Search / ZVON
 | Indexes | Syntax | Prelude | Ratio | Complex | >> Numeric << | Ix | Array | List | Maybe | Char | Monad | IO | Directory | System | Time | Locale | CPUTime | Random

Numeric

This library contains assorted numeric functions, many of which are used in the standard Prelude. Most are self-explanatory. The floatToDigits function converts a floating point value into a series of digits and an exponent of a selected base. This is used to build a set of floating point formatting functions.
```module Numeric(fromRat,
showSigned, showInt,
floatToDigits,
showEFloat, showFFloat, showGFloat, showFloat,

import Char
import Ratio
import Array

-- 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
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

showSigned    :: Real a => (a -> ShowS) -> Int -> a -> ShowS
showSigned showPos p x | x < 0 = showParen (p > 6)
(showChar '-' . showPos (-x))
| otherwise = showPos x

-- showInt is used for positive numbers only
showInt    :: Integral a => a -> ShowS
showInt n r | n < 0 = error "Numeric.showInt: can't show negative numbers"
| otherwise =
let (n',d) = quotRem n 10
r'     = toEnum (fromEnum '0' + fromIntegral d) : r
in  if n' == 0 then r' else showInt n' r'

[(-x,t) | ("-",s) <- lex r,
read'' r = [(n,s)  | (str,s) <- lex r,

-- Leading minus signs must be handled elsewhere.

readInt :: (Integral a) => a -> (Char -> Bool) -> (Char -> Int) -> ReadS a
[(foldl1 (\n d -> n * radix + d) (map (fromIntegral . digToInt) ds), r)
| (ds,r) <- nonnull isDig s ]

-- Unsigned readers for various bases

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
minExp = minExp0 - p            -- the real minimum exponent
-- 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)

-- This floating point reader uses a less restrictive syntax for floating
-- point than the Haskell lexer.  The `.' is optional.

| (ds,d) <- lexDigits r,
(ds',t) <- lexFrac d ]

lexFrac ('.':ds) = lexDigits ds
lexFrac s        = [("",s)]