#load "camlp4o.cma";;

let rec take n s =
  if n = 0 then [] else 
    match s with parser [< 'x; s >] -> x :: take (n - 1) s;;

(* 2(a) *)
let rec map f = parser 
  | [< 'x; s >] -> [< 'f x; map f s >];;

let rec from n = [< 'n; from (n + 1) >];;

let rec pi_stream_aux x =
  [< '(1.0 /. x); map (fun x -> -. x) (pi_stream_aux (x +. 2.)) >];;
  
let rec partial_sum x = parser 
  | [< 'n; s >] -> [< 'x; partial_sum (x +. n) s >];;

let pi_stream =
  map (fun x -> x *. 4.) (partial_sum 0. (pi_stream_aux 1.));;

take 10 pi_stream;;


(* 2(b) *)
let square x = x *. x;;

let euler_f x y z =
  z -. (square (z -. y) /. (x -. 2. *. y +. z));;

let rec euler_transform = parser
  | [< 'x; 'y; 'z; s >] ->
      [< 'euler_f x y z; euler_transform [< 'y; 'z; s >] >];;

take 10 (euler_transform pi_stream);;

(* 2(c) *)
let rec tableau f s =
  [< 's; tableau f (f s) >];;

let hd = parser [< 'x; s >] -> x

let accelerate f s =
  map hd (tableau f s);;

take 10 (accelerate euler_transform pi_stream);;