sicp-ex-2.13

No scheme, just math. If Ca is center of a, and Ta is tolerance of a, a is the interval

  a = [Ca*(1 - 0.5*Ta), Ca*(1 + 0.5*Ta)]

and b is

  b = [Cb*(1 - 0.5*Tb), Cb*(1 + 0.5*Tb)]

If the endpoints are positive, a*b has the endpoints (after simplifying):

  a*b = [Ca*Cb*(1 - 0.5*(Ta + Tb) + 0.25*Ta*Tb),
         Ca*Cb*(1 + 0.5*(Ta + Tb) + 0.25*Ta*Tb)]

Ta*Tb will be a wee number, so it can be ignored. So, it appears that for small tolerances, the tolerance of the product will be approximately the sum of the component tolerances.

A quick check:

  
 (define (make-interval a b) (cons a b)) 
 (define (upper-bound interval) (max (car interval) (cdr interval))) 
 (define (lower-bound interval) (min (car interval) (cdr interval))) 
 (define (center i) (/ (+ (upper-bound i) (lower-bound i)) 2)) 
  
 ;; Percent is between 0 and 100.0 
 (define (make-interval-center-percent c pct) 
   (let ((width (* c (/ pct 100.0)))) 
     (make-interval (- c width) (+ c width)))) 
  
 (define (percent-tolerance i) 
   (let ((center (/ (+ (upper-bound i) (lower-bound i)) 2.0)) 
         (width (/ (- (upper-bound i) (lower-bound i)) 2.0))) 
     (* (/ width center) 100))) 
  
 (define (mul-interval x y) 
   (let ((p1 (* (lower-bound x) (lower-bound y))) 
         (p2 (* (lower-bound x) (upper-bound y))) 
         (p3 (* (upper-bound x) (lower-bound y))) 
         (p4 (* (upper-bound x) (upper-bound y)))) 
     (make-interval (min p1 p2 p3 p4) 
                    (max p1 p2 p3 p4)))) 
  
  
 (define i (make-interval-center-percent 10 0.5)) 
 (define j (make-interval-center-percent 10 0.4)) 
 (percent-tolerance (mul-interval i j)) 
  
 ;; Gives 0.89998, pretty close to (0.5 + 0.4). 
  

My method:

  percent-interval :=> PI  old-interval :=> OI

  PI(a, p1) -> OI(a-a*p1, a+a*p1)
  PI(b, p2) -> OI(b-b*p2, b+b*p2)

  PI(c, p) = PI(a, p1)*PI(b, p2) 
    :=> OI(a-a*p1, a+a*p1) * OI(b-b*p2, b+b*p2)
         = OI((a-a*p1)*(b-b*p2), (a+a*p1)*(b+b*p2))
         = OI(cL, cU)
  c = (cL + cU)/2 
    = ((a-a*p1)*(b-b*p2) + (a+a*p1)*(b+b*p2))/2
    = a*b*(1+p1*p2)
  p = (c-cL)/c
    = (a*b*(1+p1*p2) - (a-a*p1)*(b-b*p2))/a*b*(1+p1*p2) 
    = a*b*(p1 + p2)/a*b*(1+p1*p2)
    = (p1+p2)/(1+p1*p2)

  c = a*b*(1+p1*p2);
  p = (p1+p2)/(1+p1*p2);

Example:

  PI(10, 0.005)*PI(10, 0.004) 
    = PI(10*10*(1+0.004*0.005), (0.004+0.005)/(1+0.004*0.005))
    = PI(100.002, 0.00899982);