-- Exercise 4.3
-- George C. Privon
-- 2023-12-22
type R = Double
type Derivative = (R -> R) -> R -> R

-- a function to take the numerical derivative, approximated for some dt
derivative :: R -> Derivative
derivative dt x t = (x (t + dt/2) - x (t - dt/2)) / dt

-- the function we want to take the derivative of
myfunc :: R -> R
myfunc x = x**4

-- exact derivative of our function of interest
dmyfunc :: R -> R
dmyfunc x = 4 * x**3

-- take the derivative with dt=1
dOne :: R -> R
dOne x = derivative 0.01 myfunc x

-- compare the exact derivative with the numerical derivative using dt=1
relabserr :: R -> R
relabserr x = abs (dmyfunc x - dOne x) / dmyfunc x

-- Solution
-- The derivative of f(x) = x**4, evaluated at x=0.01 with dt=0.01 produces a
-- relative error of 25%