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Problem 2.11
Friday, August 23, 2019; ago; Download .md

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))))))