(define (cube x) (* x x x))

(define (cube-root x)
  (define (improve guess)
    (/ (+ (/ x (* guess guess))
          (* 2 guess))
       3.0))
  (define epsilon 0.0001)
  (define (good-enough? guess)
    (< (abs (- (cube guess) x)) epsilon))
  (define (cube-iter guess)
    (if (good-enough? guess) guess
        (cube-iter (improve guess))))
  (if (< x 0) (- (cube-root (-x)))
      (cube-iter 1.0)))

; Test it out
(map cube-root (map cube '(0 1 2 3 4 5 6 7 8 1000 1000000 1000000000 0.001 0.000001)))
; (.03901844231062338 1. 2.000004911675504 3.0000005410641766 4.000000000076121 5.000000000287929 6.000000000047538 7.000000000000461 8.00000003355707 1000. 1000000. 1000000000. 3.9018580584466174e-2 .03901844231062351)