(load "1.35.txt") ; for fixed-point
(load "1.43.txt") ; for repeated

(define (average a b) (/ (+ a b) 2.0))

(define (average-damp f)
  (lambda (x) (average x (f x))))

(define (nth-root x n d)
  (fixed-point ((repeated average-damp d)
                   (lambda (y) (/ x
                                  (expt y (- n 1)))))
               1.0))

; Experiment: vary n
; (nth-root 2^n n d)
; nth root, d = number of average dampings required for convergence
; n d
; ---
; 2 1
; 3 1
; 4 2
; 5 2
; 6 1
; 7 2
; 8 3
; 9 3
; 10 1
; 11 3
; 12 2
; 13 3
; I do not see any clear pattern

; Experiment: vary x
; (map (lambda (x) (nth-root (* x x x x) 4 2)) (iota 100))
; (map (lambda (x) (nth-root (* x x x) 3 1)) (iota 100))