(define (average a b)
  (/ (+ a b) 2))

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

(define (cube x) (* x x x))

(define tolerance 0.00001)

(define (fixed-point f first-guess)
  (define (close-enough? v1 v2)
    (< (abs (- v1 v2)) tolerance))
  (define (try guess)
    (let ((next (f guess)))
      (if (close-enough? guess next)
          next
          (try next))))
  (try first-guess))

(define (average-damp f)
  (lambda (x) (average x (f x))))

;: ((average-damp square) 10)

;;(define (sqrt x)
;;  (fixed-point (lambda (y) (average y (/ x y)))
;;               1.0))

;;(define (sqrt x)
;;  (fixed-point (average-damp (lambda (y) (/ x y)))
;;               1.0))

(define (deriv g)
  (lambda (x)
    (/ (- (g (+ x dx)) (g x))
       dx)))
(define dx 0.00001)

(define (newton-transform g)
  (lambda (x)
    (- x (/ (g x) ((deriv g) x)))))

(define (newtons-method g guess)
  (fixed-point (newton-transform g) guess))

(define (sqrt x)
  (newtons-method (lambda (y) (- (square y) x))
                  1.0))

;;
(define (fixed-point-of-transform g transform guess)
  (fixed-point (transform g) guess))

;; exercise 1.40
(define (cubic a b c)
  (lambda (x) 
    (+ (cube x) (* a (square x)) (* b x) c)))

;; exercise 1.41
(define (inc x) 
  (display 'inc:)
  (display x)
  (newline)
  (+ x 1))

(define (double f)
  (lambda (x) 
    (display 'f:)
    (display x)
    (newline)
    (f (f x))))
;; (((double (double double)) inc) 5)