Haskell basic typeclass primer (quadratic equations)

Question

I have written Haskell code to my spec to help me grasp types and syntax, because I am learning. I hope to generalize it further to accommodate more kinds of equations. I am (perhaps prematurely) worried that repeated usage of delta in multiple places is going to slow down my code, but I don't see how I can avoid it.

I am a bit confused about whether I should use Floating class or just use Float concrete type everywhere.

I appreciate any and all tips.

Here's the expected output

Equation -10.0x^2 +4.0x +1.0 = 0 has following 2 solutions:
0.28708285
-8.708288e-2

And the code

-- MathPrimer

delta :: Floating a => a -> a -> a -> a
x1 :: Floating a => a -> a -> a -> a
x2 :: Floating a => a -> a -> a -> a

delta a b c = b * b - 4 * a * c

x1 a b c = ( - b - sqrt(delta a b c)) / ( 4 * a * c )
x2 a b c = ( - b + sqrt(delta a b c)) / ( 4 * a * c )


class Algebraic equ where
    algebraicForm :: equ -> String

class (ZeroEquation equ) where
    solve :: equ -> [Float]
    solutionCount :: equ -> Int


data Equation = Quadratic {a :: Float, b :: Float, c :: Float}
    deriving (Show, Read, Eq)

instance (ZeroEquation Equation) where
    solve (Quadratic a b c) = solutions
        where
            both = [x1 a b c, x2 a b c]
            one = [x1 a b c]
            zero = []
            solutions = if _delta < 0 then zero else if _delta == 0 then one else both
            _delta = (delta a b c)
    solutionCount (Quadratic a b c) = count
        where
            count = if _delta < 0 then 0 else if _delta == 0 then 1 else 2
            _delta = (delta a b c)

instance (Algebraic Equation) where
    algebraicForm (Quadratic a b c) = show a ++ "x^2 +" ++ show b ++ "x +" ++ show c

showEquation :: Equation -> String
showEquation equ = "Equation " ++ description ++ " = 0 has following " ++ count ++ " solutions:\n" ++ solutions
    where
        count = show $ solutionCount equ
        description = algebraicForm equ
        solutions = unlines $ map show $ solve equ

main = do
    let equation = Quadratic (-10) 4 1
    putStrLn $ showEquation equation

Answer 1

It's hard to tell if the compiler would be able to eliminate duplicate calls to delta. It may do that, by inlining the x1 and x2 functions and then eliminating common subexpressions. Personally though, I prefer to do this myself. The following modification is guaranteed to call delta only once:

solve (Quadratic a b c) = solutions
    where
        both = [x1, x2]
        one = [x1]
        zero = []
        solutions = if _delta < 0 then zero else if _delta == 0 then one else both
        _delta = (delta a b c)
        x1 = ( - b - sqrt _delta) / ( 4 * a * c )
        x2 = ( - b + sqrt _delta) / ( 4 * a * c )