SICP Exercise 2.11
The table for the min and max for each combination is as folows:
| $$L_x$$ | $$U_x$$ | $$L_y$$ | $$U_y$$ | $$L_x L_y$$ | $$L_x U_y$$ | $$U_x L_y$$ | $$U_x U_y$$ | min | max | | + | + | + | + | + | + | + | + | $$L_x L_y$$ | $$U_x U_y$$ | | + | + | - | + | - | + | + | + | $$U_x L_y$$ | $$U_x U_y$$ | | + | + | - | - | - | - | - | - | $$U_x L_y$$ | $$L_x U_y$$ | | - | + | + | + | - | - | + | + | $$L_x U_y$$ | $$U_x U_y$$ | | - | + | + | - | + | - | - | + | $$min(L_xU_y, U_x L_y)$$ | $$max (L_x L_y, U_x U_y)$$ | | - | + | - | - | + | + | - | - | $$U_x L_y$$ | $$L_x L_y$$ | | - | - | + | + | - | - | - | - | $$L_x U_y$$ | $$U_x L_y$$ | | - | - | - | + | + | - | + | - | $$L_x L_y$$ | $$L_x U_y$$ | | - | - | - | - | + | + | + | + | $$U_x U_y$$ | $$L_x L_y$$ |
So our procedure now becomes:
(define (mul-interval x y)
(define not-negative?
(lambda (x) (not (negative? x))))
(let ((lx (lower-bound x))
(ux (upper-bound x))
(ly (lower-bound y))
(uy (upper-bound y)))
(cond ((and (not-negative? lx)
(not-negative? ux)
(not-negative? ly)
(not-negative? uy)))
(make-interval (* lx ly)
(* ux uy)))
(cond ((and (not-negative? lx)
(not-negative? ux)
(negative? ly)
(not-negative? uy)))
(make-interval (* ux ly)
(* ux uy)))
(cond ((and (not-negative? lx)
(not-negative? ux)
(negative? ly)
(negative? uy)))
(make-interval (* ux ly)
(* lx uy)))
(cond ((and (negative? lx)
(not-negative? ux)
(not-negative? ly)
(not-negative? uy)))
(make-interval (* lx uy)
(* ux uy)))
(cond ((and (negative? lx)
(not-negative? ux)
(not-negative? ly)
(negative? uy)))
(make-interval (min (* lx uy) (* ux ly))
(max (* lx ly) (* ux uy)))
(cond ((and (negative? lx)
(not-negative? ux)
(negative? ly)
(negative? uy)))
(make-interval (* ux ly)
(* lx ly)))
(cond ((and (negative? lx)
(negative? ux)
(not-negative? ly)
(not-negative? uy)))
(make-interval (* lx uy)
(* ux ly)))
(cond ((and (negative? lx)
(negative? ux)
(negative? ly)
(not-negative? uy)))
(make-interval (* lx ly)
(* lx uy)))
(cond ((and (negative? lx)
(negative? ux)
(negative? ly)
(negative? uy)))
(make-interval (* ux uy)
(* lx ly))))))