1 (define tolerance 0.00001)
2 (define (fixed-point f first-guess)
3   (define (close-enough? v1 v2)
4     (< (abs (- v1 v2)) tolerance))
5   (define (try guess)
6     (let ((next (f guess)))
7       (if (close-enough? guess next)
8 	  next
9 	  (try next))))
10   (try first-guess))
11 
12 (define (average x y)
13   (/ (+ x y) 2.0))
14 (define (average-damp f)
15   (lambda (x) (average x (f x))))
16 (define (fixed-point-of-transform g transform guess)
17   (fixed-point (transform g) guess))
18 (define (sqrt x)
19   (fixed-point-of-transform (lambda (y) (/ x y))
20 			    average-damp
21 			    1.0))
22 (define (cube-root x)
23   (fixed-point-of-transform (lambda (y) (/ x (square y)))
24 			    average-damp
25 			    1.0))
26 
27 (define (compose f g)
28   (lambda (x)
29     (f (g x))))
30 
31 (define (test-case actual expected)
32   (load-option 'format)
33   (newline)
34   (format #t "Actual: ~A Expected: ~A" actual expected))
35 
36 (define (square x) (* x x))
37 (define (inc x) (1+ x))
38 ;; (test-case ((compose square inc) 6) 49)
39 
40 (define (repeated f n)
41   (if (= n 0)
42       (lambda (x) x)
43       (compose f (repeated f (- n 1)))))
44 
45 ;; (test-case ((repeated square 2) 5) 625)
46 (test-case (cube-root 5) 1.70997594668)
47 
48 
49 ;; Exercise 1.44.  The idea of smoothing a function is an important concept in signal processing. If f is a function and dx is some small number, then the smoothed version of f is the function whose value at a point x is the average of f(x - dx), f(x), and f(x + dx). Write a procedure smooth that takes as input a procedure that computes f and returns a procedure that computes the smoothed f. It is sometimes valuable to repeatedly smooth a function (that is, smooth the smoothed function, and so on) to obtained the n-fold smoothed function. Show how to generate the n-fold smoothed function of any given function using smooth and repeated from exercise 1.43. 
50 
51 (define dx 0.01)
52 
53 (define (smooth f)
54   (lambda (x)
55     (/ (+ (f x) 
56 	  (f (+ x dx))
57 	  (f (- x dx)))
58        3.0)))
59 (define (n-fold-smooth f n)
60   ((repeated smooth n) f))