{-# LANGUAGE FlexibleInstances #-}

module RealFunction (
  RealFunction,
  cmult,
  fexp
  )
where


type RealFunction a = (a -> Double)


-- | A 'Show' instance is required to be a 'Num' instance.
instance Show (RealFunction a) where
    -- | There is nothing of value that we can display about a
    --   function, so simply print its type.
    show _ = "<RealFunction>"


-- | An 'Eq' instance is required to be a 'Num' instance.
instance Eq (RealFunction a) where
    -- | Nothing else makes sense here; always return 'False'.
    _ == _ = error "You can't compare functions for equality."


-- | The 'Num' instance for RealFunction allows us to perform
--   arithmetic on functions in the usual way.
instance Num (RealFunction a) where
    (f1 + f2)  x = (f1 x) + (f2 x)
    (f1 - f2)  x = (f1 x) - (f2 x)
    (f1 * f2)  x = (f1 x) * (f2 x)
    (negate f) x = -1 * (f x)
    (abs    f) x = abs (f x)
    (signum f) x = signum (f x)
    fromInteger i _ = fromInteger i


-- | Takes a constant, and a function as arguments. Returns a new
--   function representing the original function times the constant.
cmult :: Double -> (RealFunction a) -> (RealFunction a)
cmult coeff f = (*coeff) . f


-- | Takes a function f and an exponent n. Returns a new function,
--   f^n.
fexp :: (RealFunction a) -> Int -> (RealFunction a)
fexp f n
     | n == 0 = (\_ -> 1)
     | otherwise = \x -> (f x)^n