sicp exercise 1.31

;  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 pie using the formula
;
;  pi/4 = 2/3.4/3.4/5.6/5.6/7 ....
;
;  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.

(define (prod-recur term a next b)
    (if (> a b)
        1
        (* (term a)
           (prod-recur term (next a) next b))))

(define (prod-iter term a next b)
    (define (prod-iter-impl term a next b res)
        (if (> a b)
            res
            (prod-iter-impl term (next a) next b (* (term a) res))))
    (prod-iter-impl term a next b 1))

(define (fact n)
   (define (next x) (+ x 1))
   (define (term x) x)
   (prod-iter term 1 next n))

;(display (fact 500)) (newline)

(define (pi)
   (define (next x) (+ x 1))
   (define (term x)
       (cond ((even? x) (/ x (+ x 1)))
             (else (/ (+ x 1) x))))
   (* (prod-iter term 2 next 1000) 4))

(display (rationalize (pi) 0.000001)) (newline)