(* file: bisection.ml
   author: Bob Muller

   CS1103 Computer Science I Honors

   Topics:

   - The Babylonian Bisection Algorithm for computing an approximation
     of the square root of a (possibly irrational) number.

  babylonianSqrt : float -> int -> float -> float
*)
let babylonianSqrt x n error =
  let closeEnough a b = abs_float (a -. b) < error in
  let rec repeat i lo hi =
    match i > n with
    | true  -> failwith "unable to compute acceptable approximation"
    | false ->
      let middle = (lo +. hi) /. 2.0 in
      let midSquared = middle ** 2.0
      in
      Printf.printf "attempt=%d: lo=%f, hi=%f, mid=%f\n" i lo hi middle ;
      match closeEnough midSquared x with
      | true  -> middle
      | false ->
        (match midSquared < x with
         | true  -> repeat (i + 1) middle hi
         | false -> repeat (i + 1) lo middle)
  in
  repeat 1 0.0 x

(* The following version has a slightly more convenient interface.
   And it covers material not covered in lecture. In particular,
   the definition has default values for the number of iterations
   and the error.  NB: the type.

   babylonianSqrt : ?n:int -> ?error:float -> float -> float
*)
let babylonianSqrt ?n:(n = 30) ?error:(error = 1e-6) x =
  let closeEnough a b = abs_float (a -. b) < error in
  let rec repeat i lo hi =
    match i > n with
    | true  -> failwith "unable to compute acceptable approximation"
    | false ->
      let middle = (lo +. hi) /. 2.0 in
      let midSquared = middle ** 2.0
      in
      Printf.printf "attempt=%d: lo=%f, hi=%f, mid=%f\n" i lo hi middle ;
      match closeEnough midSquared x with
      | true  -> middle
      | false ->
        (match midSquared < x with
         | true  -> repeat (i + 1) middle hi
         | false -> repeat (i + 1) lo middle)
  in
  repeat 1 0.0 x