SATySFi：サイン・コサインを使って平方根を求める

squareroot.saty

@require: stdja
@require: list

% sqrt r = sin(acos(1 - 2r)/2)   (for 0 < r < 1)
let sqrt-b r =
  sin (acos (1. -. r *. 2.) /. 2.)

% NOTE: behavior undefined
let get-invalid-value () = 0. /. 0.

let sqrt r =
  let rd = 1pt *' r in
    if rd >' 0pt then
      if rd >' 1pt then 1. /. sqrt-b (1. /. r)
      else sqrt-b r
    else if rd <' 0pt then get-invalid-value ()
    else 0.

%===========================================================

let my-iota n =
  let-rec iter | 0 rs = rs
               | k rs = iter (k - 1) ((k - 1) :: rs) in
    iter n []

let my-sqrt-eqn n =
  let rn = sqrt (float n) in
  let mthn = math-char MathOrd (arabic n) in
  let mthrn = math-char MathOrd (show-float rn) in
  ${| \sqrt{#mthn} |= #mthrn |}

let my-sqrt-eqns n =
  List.map (fun k -> my-sqrt-eqn (k + 1)) (my-iota n)

in

document (|
  title = {平方根の値}; author = {某ZR（アレ）};
  show-title = true; show-toc = false;
|) '<
  +p {
    サイン・コサインは\SATySFi;で平方根を求めるのに非常に役に立つ。
    \align(my-sqrt-eqns 25);
  }
>