SICP Exercise 2.12: Tolerating Engineers

After debugging her program, Alyssa shows it to a potential user, who complains that her program solves the wrong problem. He wants a program that can deal with numbers represented as a center value and an additive tolerance; for example, he wants to work with intervals such as 3.5± 0.15 rather than [3.35, 3.65]. Alyssa returns to her desk and fixes this problem by supplying an alternate constructor and alternate selectors:

(define (make-center-width c w)
  (make-interval (- c w) (+ c w)))
(define (center i)
  (/ (+ (lower-bound i) (upper-bound i)) 2))
(define (width i)
  (/ (- (upper-bound i) (lower-bound i)) 2))

Unfortunately, most of Alyssa's users are engineers. Real engineering situations usually involve measurements with only a small uncertainty, measured as the ratio of the width of the interval to the midpoint of the interval. Engineers usually specify percentage tolerances on the parameters of devices, as in the resistor specifications given earlier.

Define a constructor make-center-percent that takes a center and a percentage tolerance and produces the desired interval. You must also define a selector percent that produces the percentage tolerance for a given interval. The center selector is the same as the one shown above.

The percentage tolerance is another way of specifying the width of an interval, so our constructor will share some commonality with make-center-width. It will take two parameters, the first of which is the center value of the interval. It will also call make-interval with two bounds calculated by respectively subtracting and adding the width of the interval from the center value. Where it will differ will be in the second operand and what we do with it. The second parameter will express the width of the interval as a percentage of the center value, rather than as an absolute value. We will then have to use this to calculate the width of the interval so that we can use that to calculate the lower and upper bounds of the interval. This is simply a case of multiplying the center value by the percentage value and then dividing by 100.

Here's our constructor:

(define (make-center-percent c p)
  (let ((w (/ (* c p) 100)))
    (make-interval (- c w) (+ c w))))

To define the selector percent we need to find the width of the interval and then determine what percentage of the center value the width is. I.e. reverse the calculation we used above to produce the width from the percentage tolerance. Given that we have the center and width selectors Alyssa defined above we can define our selector simply as:

(define (percent i)
  (/ (* (width i) 100) (center i)))

Alternatively, if we express this as an algebraic expression it's easy to see how we could reduce the operations in the calculation. If we express the lower bound of the interval as l_i and the upper bound as u_i then the calculation of (percent i) is equivalent to:

  100 × (ui - li)/2
     (li + ui)/2
= 100 × (ui - li)/2
        (li + ui)/2
= 100 × ui - li
        li + ui

So we could also define percent as:

(define (percent i)
  (* 100
     (/ (- (upper-bound i) (lower-bound i))
        (+ (upper-bound i) (lower-bound i)))))

This removes the divide-by-two that occurs in the calculations of both the width and the center values, only to have the division of width by center values to cancel out the divide-by-twos.

Let's test it out to confirm it works:

> (define cp1 (make-center-percent 100 10))
> (define cp2 (make-center-percent 50 25))
> (center cp1)
100
> (center cp2)
50
> (percent cp1)
10
> (percent cp2)
25