Exercise 1.31

a. The sum procedure is only the simplest of a vast number of similar abstractions that can be captured as higher-order procedures.51 Write an analogous procedure called product that returns the product of the values of a function at points over a given range. Show how to define factorial in terms of product. Also use product to compute approximations to using the pi / 4 = (2/3) * (4/3) * (4/5) * (6/5) * (6/7) * (8/7) ...

b. If your product procedure generates a recursive process, write one that generates an iterative process. If it generates an iterative process, write one that generates a recursive process.

I wrote the following:


(define (product term a next b)
    ((> a b) 1)
    (else (* (term a) (product term (next a) next b)))))


(define (i-product term a next b)
  (cond ((> a b) null)
       (define (iter a result)
         (cond ((> a b) result)
               (else (iter (next a) (* (term a) result)))))
       (iter a 1))))

Multiply-integers [test - does (product ...) work?]

(define (identity x) x)
(define (inc x) (+ 1 x))
(define (multiply-integers a b) (i-product identity a inc b))

Compute pi:

(define (square x) (* x x))
(define (compute-pi steps)
  (define (next n) (+ n 2.0))
  (* 8.0 (* steps 2) (/ (i-product square 4.0 next (* (- steps 1) 2)) 
              (i-product square 3.0 next (* steps 2)))))


(define (factorial n)
  (define (next n) (+ n 1))
  (i-product identity 1 next n))

What do you think of my solution?

  • \$\begingroup\$ I meant to post this at codereview.stackexchange.com - can it be migrated? \$\endgroup\$
    – jaresty
    Mar 30, 2011 at 2:52
  • \$\begingroup\$ @Joshua: Yes. Click "flag", "it needs ♦ moderator attention", then "other", and fill in the textbox requesting this. :-) (I've also requested a migration for you in the mod chat room, in case they monitor that more frequently. ;-)) \$\endgroup\$ Mar 30, 2011 at 3:07
  • \$\begingroup\$ It turned out that the mod chat room was indeed monitored more frequently (thanks Shog9!), but, in future, you can flag your post yourself anyway. SO mods are meant to be pretty active, and should see your flag pretty much straight away. :-) \$\endgroup\$ Mar 30, 2011 at 3:47

1 Answer 1


Since your definitions have only two cond clauses, you may replace cond with if. Your iterative definition is a departure from the example of sum. It should return 1 (product identity) when a is greater than b, not null. This makes the outer cond unnecessary.

Your definition of compute-pi suffers from imprecision of float operations (it fails to produce meaningful values for n > 85). It would be better to convert to float after computing the approximating fraction.

(define (product term a next b)
  (if (> a b)
      (* (term a) (product term (next a) next b))))

(define (i-product term a next b)
  (define (iter a result)
    (if (> a b)
        (iter (next a) (* (term a) result))))
  (iter a 1))

(define (compute-pi steps)
  (define (next n) (+ n 2))
  (* 8.0 (* steps 2) (/ (i-product square 4 next (* (- steps 1) 2)) 
                        (i-product square 3 next (* steps 2)))))

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