\$ cd ~

# 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))))))