# SICP - Solution: Exercise 1.8

## Sep 30, 2018 21:03 · 137 words · 1 minute read

Exercise 1.8: Newton’s method for cube roots is based on the fact that if $y$ is an approximation to the cube root of $x$, then a better approximation is given by the value

$${\frac{{x/y^2}+2y}3.}$$

Use this formula to implement a cube-root procedure analogous to the square-root procedure. (In 1.3.4 we will see how to implement Newton’s method in general as an abstraction of these square-root and cube-root procedures.)

Solution

Using the same improvement for large and small number from exercise 1.7:

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

(define (good-enough? previous-guess guess)
(< (abs (/ (- guess previous-guess) guess)) 0.00000000001))

(define (cube-root-iter guess x)
(if (good-enough? (improve guess x) guess)
guess
(cube-root-iter (improve guess x) x)))

(define (improve guess x)
(/ (+ (/ x (* guess guess)) (* 2 guess)) 3))

(define (cube-root x)
(cube-root-iter 1.0 x))