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
(/ (+ (/ x (squre y))
		  (* y 2))
	 3)
	 
(x/(y^2)+2y)/3
Use this formula to implement a cube-root procedure analogous to the square-root procedure. (In
section 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.)



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


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




(defun improve_c (guess x)
  (/ (+ (/ x (squre guess))
	(* 2 guess))
     3))


(defun good-enough_c (guess x)
  (< (abs (/ (- guess (improve_c guess x))
	     guess
	     ))
     0.0000001))


(defun cbrt-iter-ex (guess x)
  (if (good-enough_c guess x)
      guess
      (cbrt-iter-ex  (improve_c guess x)
		     x)))


(defun cuberoot(x)
  (cbrt-iter-ex 1.0 x))

CL-USER> (cuberoot (* 64.0 8))
8.0