Taylor

// Kolosova Alexandra
// variant #5

let a = 0.0
let b = 0.5
let n = 10
let eps = 0.00000001

let iter f i a b =
    let rec iter' a acc =
        if a > b then acc
        else iter' (a+1) (f a acc)
    iter' a i

// exponentiation
//let power x = iter (fun _ acc -> acc * x) 1. 1

// 2 * ( cos^2 (x) - 1 )

//let f (x:float) = cos >> power 2 >> (-) 1. >> (*) 2.
//let f x = 2. * ((power (cos(x)) 2) - 1.)
let f (x:float) = 2. * (float(cos(x)**2.) - 1.)

// (-1)^n * (2x)^2n / ((2n)!)

let fact =
    let rec fact' acc = function
       | 1 -> acc
       | n -> fact' (acc*n) (n-1)
    fact' 1


// calculate i element
//let taylorElement x i = (0. - power x n)* power (2.*x) (2 * n) / float((fact(2*n) ))
let taylorElement (x:float) n = (-1.0)**float(n) * (float (2.*x)**float (2*n)) / float(fact(2*n))

// Taylor series in "dumb" way:
let taylorNaive x = iter (fun n acc -> acc + ((-1.)** float n) * (2. * x)**(2. * float n) / float((fact(2*n) ))) 0. 0 n
//let taylorNaive x = iter (fun n acc -> acc + taylorElement x n) 0. 0 n

// Taylor series in a more efficient way
let taylor (x:float) =
    let rec loop prev i acc =
        if (abs(taylorElement x i ) < eps) then acc
        else loop (taylorElement x i) (i + 1) (acc + taylorElement x i)
    loop (-2.*x*x) 1 0.


let main =
  for i=0 to n do
    let x = a + (float i) / (float n) * (b - a)
    printfn "%5.2f  %10.6f  %10.6f   %10.6f" x (f x) (taylorNaive x) (taylor x)

main