<jupyter_start><jupyter_text>**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.)<jupyter_code>(define (good-enough? previous-guess guess)
  (< (abs (/ (- guess previous-guess) guess)) 0.001))

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

(define (cubic-root x)
  (define (iter guess x)
    (if (good-enough? guess x)
        guess
        (iter (improve guess x) x)))
  (iter 1.0 x))
(cubic-root 27)<jupyter_output><empty_output>
{'fraction_non_alphanumeric': 0.13493975903614458, 'fraction_numerical': 0.021686746987951807, 'mean_word_length': 3.9760479041916166, 'pattern_counts': {'":': 0, '<': 6, '<?xml version=': 0, '>': 5, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'chain_length': '1', 'dir': 'jupyter-structured-clean-dedup', 'id': '397744', 'license': 'Unlicense', 'n_tokens_mistral': 297, 'n_tokens_neox': 267, 'n_words': 105, 'path': '1/8.ipynb', 'repo_name': 'BasixKOR/SICP', 'size': '830'}