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?



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

(defun improve (guess x)
  (average (/ x guess)
	   guess))

(defun squre(x)
  (* x x))

(defun good-enough? (guess x)
    (< (abs (- (squre guess)
	        x))
     0.0000001))

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

(defun mysqrt(x)
  (sqrt-iter 1.0 x))




(defun good-enough_ex(guess x)
  (< (abs (/ (- guess (improve guess x))
	     guess
	     ))
     0.0000001))


(defun sqrt-iter-ex(guess x)
  (if (good-enough_ex guess x)
      guess
      (sqrt-iter-ex (improve guess x)
		    x)))

(defun mysqrt_ex(x)
  (sqrt-iter-ex 1.0 x))


CL-USER> (mysqrt_ex 0.0000000009)
3.e-5
CL-USER> (mysqrt 0.0000000009)
2.453682e-4
CL-USER> 
CL-USER> (mysqrt_ex 99999999999999999999999999999999999999)
1.e19
CL-USER> (mysqrt    99999999999999999999999999999999999999)
; Evaluation aborted on #.
CL-USER> 

1,when the number is too small, in good-enough function, the squre function will make a more  inaccurate result, because the float encode method in computer science is inaccurate .
  

2,when the number is too large, in good-enough function, the squre function will make a much larer number, which the computer can not store and cause overflow. but the  good-enough_ex function 
  make a small number.