a. The filtered_accumulate procedure is not optimized at all.. We'll use the prime? procedure defined earlier in the book with a small fix to make (prime? 1) -> #f.

  (define (smallest-div n) 
    (define (divides? a b) 
      (= 0 (remainder b a))) 
    (define (find-div n test) 
       (cond ((> (sq test) n) n) ((divides? test n) test) 
             (else (find-div n (+ test 1))))) 
    (find-div n 2)) 
  
   (define (prime? n) 
     (if (= n 1) false (= n (smallest-div n))))

Filtered accumulate procedure:

  (define (filtered-accumulate combiner null-value term a next b filter) 
  (if (> a b) null-value 
      (if (filter a) 
          (combiner (term a) (filtered-accumulate combiner null-value term (next a) next b filter)) 
          (combiner null-value (filtered-accumulate combiner null-value term (next a) next b filter)))))

A slightly dryer recursive version:

 (define (filtered-accumulator filter combiner null-value term a next b) 
   (if (> a b) null-value 
     (combiner (if (filter a) (term a) 
                 null-value) 
               (filtered-accumulator filter combiner null-value term (next a) next b))))

And an iterative version:

 (define (filtered-accumulate combiner null-value term a next b filter) 
   (define (iter a result) 
     (cond ((> a b) result) 
           ((filter a) (iter (next a) (combiner result (term a)))) 
           (else (iter (next a) result)))) 
   (iter a null-value))

a. Sum of squares of prime numbers procedure:

  (define (sum-of-prime-squares a b) (filtered-accumulate + 0 sq a inc b prime?))

Test example: (sum-of-prime-squares 1 5)

ans. 38

b. Product of all positive integers les than n that are relatively prime to n

 (define (gcd m n) 
   (cond ((< m n) (gcd n m)) 
         ((= n 0) m) 
         (else (gcd n (remainder m n))))) 
  
 (define (relative-prime? m n) 
 (= (gcd m n) 1)) 
  
 (define (product-of-relative-primes n) 
   (define (filter x) 
     (relative-prime? x n)) 
 (filtered-accumulate * 1 identity 1 inc n filter))

Example: (product-of-relative-primes 10) 189

poly

the solutions above:

  
 (define (filtered-accumulate combiner null-value term a next b filter) 
  (if (> a b) null-value 
      (if (filter a) 
          (combiner (term a) (filtered-accumulate combiner null-value term (next a) next b filter)) 
          (combiner null-value (filtered-accumulate combiner null-value term (next a) next b filter)))))

There is no need to combine null-value with the later accumulating actually.

 (define (filtered-accumulate filter combiner null-value term a next b) 
   (if (> a b) 
       null-value 
       (if (filter a) 
           (combiner (term a) 
                     (filtered-accumulate filter combiner null-value term (next a) next b)) 
           (filtered-accumulate filter combiner null-value term (next a) next b)))) 
  
 ; so the accumulate can be: 
 (define (accumulate combiner null-value term a next b) 
   (filtered-accumulate (lambda (x) true) 
                        combiner null-value term a next b))

BTW: I can't stand the indent of the above guy's codes.

cgb

a way to avoid code repetition both in the recursive and in the iterative version is to add a conditional in the first term of combiner. This way there is no need to call two versions of filtered-accumulate. The recursive version will be:

 (define (filtered-accumulate combiner null-value term a next b filter) 
   (if (> a b) 
       null-value 
       (combiner (if (filter a) 
                     (term a) 
                     null-value) 
                 (filtered-accumulate combiner null-value term (next a) next b filter))))

And the iterative version:

 (define (filtered-accumulate combiner null-value term a next b filter) 
   (define (iter a result) 
     (if (> a b) 
         result 
         (iter (next a) (combiner result 
                                  (if (filter a) 
                                      (term a) 
                                      null-value))))) 
   (iter a null-value))

ctz

An alternative way making use of the accumulate procedure before:

  
 (define (filtered-accumulate filter comb null-val term a next b) 
   (define (filtered-term k) 
     (if (filter k) (term k) null-val)) 
   (accumulate comb null-val filtered-term a next b))

@poly: I don't see the point of defining accumulate in terms of filtered-accumulate.

sicp-ex-1.33