Haskell: Line of best-fit

module Line where

data Line = Line Double Double

instance Show Line where
    show (Line 0 b) = "y = " ++ show b
    show (Line a b) = "y = " ++ show a ++ "x" ++ case signum b of
        0 -> ""
        1 -> " + " ++ show b
        _ -> " - " ++ show (-b)

type Point = (Double, Double)
type Equation = (Double, Double, Double)

bestFitLine :: [Point] -> Line
bestFitLine pts = uncurry Line $ solveSystem (reduce derivA) (reduce derivB)
  where
    derivA = map (\(x, y) -> mapThd (*x) (x, 1, -y)) pts
    derivB = map (\(x, y) -> (x, 1, -y)) pts

solveSystem :: Equation -> Equation -> (Double, Double)
solveSystem (a, b, c) (d, e, f) =
    let x = (b * f - e * c) / (a * e - b * d)
     in (x, (a * x + c) / (-b))

reduce :: [Equation] -> Equation
reduce = mapThd sum . unzip3

mapThd :: (a -> b) -> (a, a, a) -> (b, b, b)
mapThd f (a, b, c) = (f a, f b, f c)