1 (define (add-interval x y)
2   (make-interval (+ (lower-bound x) (lower-bound y))
3 		 (+ (upper-bound x) (upper-bound y))))
4 ;; (define (div-interval x y)
5 ;;   (mul-interval x
6 ;; 		(make-interval (/ 1.0 (upper-bound y))
7 ;; 			       (/ 1.0 (lower-bound y)))))
8 
9 (define (make-interval lower upper)
10   (cons lower upper))
11 (define (upper-bound interval)
12   (cdr interval))
13 (define (lower-bound interval)
14   (car interval))
15 
16 (define (sub-interval x y)
17   (make-interval (- (lower-bound x) (upper-bound y))
18 		 (- (upper-bound x) (lower-bound y))))
19 
20 (define (div-interval x y)
21   (define (spans-zero? interval)
22     (and (<= (lower-bound interval) 0)
23 	 (<= 0 (upper-bound interval))))
24   (if (spans-zero? y)
25       (error "Division by zero")
26       (mul-interval x
27 		    (make-interval (/ 1.0 (upper-bound y))
28 				   (/ 1.0 (lower-bound y))))))
29 
30 ;; Exercise 2.11.  In passing, Ben also cryptically comments: ``By testing the signs of the endpoints of the intervals, it is possible to break mul-interval into nine cases, only one of which requires more than two multiplications.'' Rewrite this procedure using Ben's suggestion. 
31 
32 (define (mul-interval x y)
33   (let ((p1 (* (lower-bound x) (lower-bound y)))
34 	(p2 (* (lower-bound x) (upper-bound y)))
35 	(p3 (* (upper-bound x) (lower-bound y)))
36 	(p4 (* (upper-bound x) (upper-bound y))))
37     (make-interval (min p1 p2 p3 p4)
38 		   (max p1 p2 p3 p4))))
39 
40 (define (mul-interval x y)
41   (let ((lx (lower-bound x))
42 	(ly (lower-bound y))
43 	(ux (upper-bound x))
44 	(uy (upper-bound y)))
45     (cond ((and (< ux 0) 
46 		(< uy 0)) (make-interval (* ux uy)
47 					 (* lx ly)))
48 	  ((and (> lx 0)
49 		(> ly 0)) (make-interval (* lx ly)
50 					 (* ux uy)))
51 	  ((and (< ux 0)
52 		(> ly 0)) (make-interval (* lx uy)
53 					 (* ux ly)))
54 	  ((and (> lx 0)
55 		(< uy 0)) (make-interval (* ux ly)
56 					 (* lx uy)))
57 	  ((and (< lx 0)
58 		(> ux 0)
59 		(< uy 0)) (make-interval (* ux ly)
60 					 (* lx ly)))
61 	  ((and (< lx 0)
62 		(> ux 0)
63 		(> ly 0)) (make-interval (* lx uy)
64 					 (* ux uy)))
65 	  ((and (< ux 0)
66 		(< ly 0)
67 		(> uy 0)) (make-interval (* lx uy)
68 					 (* lx ly)))
69 	  ((and (> lx 0)
70 		(< ly 0)
71 		(> uy 0)) (make-interval (* ux ly)
72 					 (* ux uy)))
73 	  ((and (< lx 0)
74 		(> ux 0)
75 		(< ly 0)
76 		(> uy 0)) (make-interval (min (* lx uy)
77 					      (* ux ly))
78 					 (max (* lx lx)
79 					      (* ux uy)))))))
80