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.

This exercise essentially requires us to rewrite the improve implementation from the previous exercise for cube roots:

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

Copying the rest with few modifications, we get:

(define (cbrt x)
  (cbrt-iter 1.0 (improve 1.0 x) x))

(define (cbrt-iter guess new-guess x)
  (if (good-enough? guess new-guess)
      guess
      (cbrt-iter new-guess (improve new-guess x) x)))

(define (good-enough? guess new-guess)
  (< (/ (abs (- new-guess guess)) guess) 0.0001))

Trying cbrt out:

(cbrt 8)

2.000004911675504

(cbrt 343)

7.000001795382107

(cbrt 0.33)

0.6910423430026196

(cbrt -8)

-2.0