The task is to write a function
findFraction maxDen number to find a short but good rational approximation to a decimal representation of a number — problem statement on TopCoder:
Given a fraction F = A/B, where 0 <= A < B, its quality of approximation with respect to number is calculated as follows:
- Let S be the decimal fraction (infinite or finite) representation of F.
- Let N be the number of digits after the decimal point in number. If number has trailing zeros, all of them are considered to be significant and are counted towards N.
- If S is infinite or the number of digits after the decimal point in S is greater than N, only consider the first N decimals after the decimal point in S. Truncate the rest of the digits without performing any kind of rounding.
- If the number of digits after the decimal point in S is less than N, append trailing zeroes to the right side until there are exactly N digits after the decimal point.
- The quality of approximation is the number of digits in the longest common prefix of S and number. The longest common prefix of two numbers is the longest string which is a prefix of the decimal representations of both numbers with no extra leading zeroes. For example, “3.14” is the longest common prefix of 3.1428 and 3.1415.
[…] You are only allowed to use fractions where the denominator is less than or equal to
maxDen. Find an approximation F = A/B of number such that 1 <= B <=
maxDen, 0 <= A < B, and the quality of approximation is maximized. Return a String formatted “A/B has X exact digits” (quotes for clarity) where A/B is the approximation you have found and X is its quality. If there are several such approximations, choose the one with the smallest denominator among all of them. If there is still a tie, choose the one among those with the smallest numerator.
import Data.Char import Data.List import Data.Maybe showResult :: (Int, Int, Int) -> String showResult (a, b, x) = show a ++ "/" ++ show b ++ " has " ++ show x ++ " exact digits" compareDigits :: [Int] -> [Int] -> Ordering compareDigits xs  = EQ compareDigits (x:xs) (y:ys) = case compare x y of EQ -> compareDigits xs ys order -> order toDigit :: Char -> Int toDigit c = ord c - ord '0' preciseDivision :: Int -> Int -> [Int] preciseDivision a b = div a' b : preciseDivision (mod a' b) b where a' = 10 * a estimate :: [Int] -> Double estimate = sum . zipWith (flip (/)) (iterate (*10) 10) . map fromIntegral bestNumerator :: [Int] -> Double -> Int -> Maybe Int bestNumerator ds p b = case find atLeast . map (pair ds b) $ [pivot .. ] of Just (a, EQ) -> Just a otherwise -> Nothing where pivot = truncate $ p * fromIntegral b pair ds b a = (a, compareDigits (preciseDivision a b) ds) atLeast (a, order) = order /= LT bestResult :: Int -> [Int] -> Maybe (Int, Int, Int) bestResult n ds = fromMaybe Nothing . find isJust . map (toResult ds $ estimate ds) $ [1..n] where toResult ds p b = case bestNumerator ds p b of Just a -> Just (a, b, 1 + length ds) Nothing -> Nothing findFraction' :: Int -> [Int] -> (Int, Int, Int) findFraction' n = fromJust . last . takeWhile isJust . map (bestResult n) . inits findFraction :: Int -> String -> String findFraction n = showResult . findFraction' n . map toDigit . tail . tail
I’m most worried by
bestResult here. I’m using a
find idiom that I invented myself, and I wonder if cleaner, more Haskelly alternatives exist.
Hmmm, not much ideas, only syntax:
showResult :: (Int, Int, Int) -> String showResult (a, b, x) = concat [show a, "/", show b, " has ", show x, " exact digits"] preciseDivision :: Int -> Int -> [Int] preciseDivision a b = let (d,r) = divMod (10*a) b in d : preciseDivision r b ...where toResult ds p b = fmap (a -> (a, b, 1 + length ds)) $ bestNumerator ds p b
Ordering is an instance of
import Data.Monoid compareDigits :: [Int] -> [Int] -> Ordering compareDigits _  = EQ compareDigits (x:xs) (y:ys) = compare x y `mappend` compareDigits xs ys
toDigit is not exactly necessary;
digitToInt, which does about the same thing. The difference is that it supports hex numbers and will fail if the character is not a valid hex digit, whereas your function assumes a decimal digit, is more efficient, but will silently give incorrect results for non-digits.
Whether you replace it or not, though,
toDigit is backwards, since the function takes a digit and gives you the numeric value.
pair, you shadow the
b parameters. That confuses me. In
toResult also has a shadowing
In general, I think many of your parameter names are too short. I realize that this is common in Haskell, but I still find it abhorrent for things that aren’t abstract. For example,
n instead of
maxDen, as in the problem statement, but “den” is an obscure abbreviation).