sicp exercise 1.7

;  Exercise 1.7.  The good-enough? test used in computing square roots will not be very effective for
;  finding the square roots of very small numbers. Also, in real computers, arithmetic operations are almost
;  always performed with limited precision. This makes our test inadequate for very large numbers. Explain
;  these statements, with examples showing how the test fails for small and large numbers. An alternative
;  strategy for implementing good-enough? is to watch how guess changes from one iteration to the next and
;  to stop when the change is a very small fraction of the guess. Design a square-root procedure that uses
;  this kind of end test. Does this work better for small and large numbers?


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

(define (improve guess x)
  (average guess (/ x guess)))


(define (average x y)
  (/ (+ x y) 2))

(define (square x) (* x x))

(define (good-enough? guess x)
  (< (/ (abs (- (square guess) x)) x) 0.001))

(define (sqrt x)
  (sqrt-iter 1.0 x))

(display (sqrt 4)) (newline)
(display (sqrt 0.0004)) (newline)
; This works better for small numbers