;;; random-project.ss: How often must fractions be reduced?

;;; John David Stone
;;; Department of Mathematics and Computer Science
;;; Grinnell College
;;; stone@math.grin.edu

;;; Original version: February 7 -- 9, 1997

;; When Chez Scheme constructs an exact rational number, it always reduces
;; the fraction to lowest terms immediately.  (For instance, if you type in
;; 100/75, Chez Scheme will immediately reduce the fraction to 4/3 and
;; store the value in that form.)
;;
;; In many cases, of course, the fraction will already be so reduced; if
;; you type in 5/7, the number is stored as 5/7, and that's that.  This
;; suggests a question:  How often -- in what proportion of the cases --
;; will a reduction be necessary?  To make it a little more exact, we could
;; choose some enormous upper bound (one trillion, for instance) --

(define enormous-upper-bound 1000000000000)

;; -- and consider all of the fractions with numerators and denominators in
;; the range from 0 to this upper bound.  (Actually, we shouldn't allow the
;; denominator to be 0, but apart from that we'll allow both the numerator
;; and the denominator to range over this enormous interval.)  How many of
;; these fractions can be reduced? 
;;
;; It would take too long to type in each of these fractions or even to
;; have Scheme generate them all; there are nearly a septillion of them.
;; But if we're satisfied with an approximate answer we can construct a
;; reliable estimate by taking a moderately large random sample of the
;; fractions and determining how many of the fractions in the sample must
;; be reduced.
;;
;; A random numerator would be any non-negative integer less than the upper
;; bound plus one:

(define choose-random-numerator
  (lambda ()
    (random (+ 1 enormous-upper-bound))))

;; A random denominator would be any positive integer less than that upper
;; bound plus one.  To ensure a positive result, we can use RANDOM to give
;; us a value one unit smaller than the one we want, and then add 1 to it:

(define choose-random-denominator
  (lambda ()
    ; Complete this procedure.
    ))

;; Notice the difference between these two definitions:
;; CHOOSE-RANDOM-NUMERATOR can return 0, CHOOSE-RANDOM-DENOMINATOR cannot.

;; A fraction can be reduced if, and only if, its numerator and its
;; denominator have a common divisor greater than 1.  Scheme's built-in
;; GCD function can therefore be put to good use in this project:

(define reducible?
  (lambda (numer denom)
    (< 1 (gcd numer denom))))

;; A call such as (REDUCIBLE? 100 75) yields #T, because the greatest
;; common divisor of 100 and 75 is 25, and 1 is less than 25.  On the other
;; hand, (REDUCIBLE? 5 7) is #F, because the greatest common divisor of
;; 5 and 7 is 1, and 1 is not less than itself.

;; We can construct and test a random fraction from among nearly a
;; septillion possibilities by invoking CHOOSE-RANDOM-NUMERATOR and
;; CHOOSE-RANDOM-DENOMINATOR to get the arguments for REDUCIBLE?:

(define test-random-fraction
  (lambda ()
    ; Complete this procedure.
    ))

;; Given a non-negative integer TEST-COUNT, the MULTIPLE-TEST procedure
;; invokes TEST-RANDOM-FRACTION the specified number of times and reports
;; the number of cases in which the fraction was reducible:

(define multiple-test
  (lambda (test-count)
    (cond ((zero? test-count) 0)
          ; Complete this procedure.
          )))

;; Given the same non-negative integer TEST-COUNT, the REDUCING-RATE
;; procedure computes the ratio between the number of reducible fractions
;; generated and the total number of test cases.

(define reducing-rate
  (lambda (test-count)
    (/ (multiple-test test-count) test-count)))

;; Theory says that values returned by this last procedure should be
;; approximately equal to 1 - 6/pi^2, which is about 0.39207.  Theory
;; in this case is personified by G. Lejeune Dirichlet, who proved the
;; mathematical version of this result in 1849, in a paper cited by
;; Donald E. Knuth on p. 324 of his _Seminumerical algorithms_ (Reading,
;; Massachusetts: Addison-Wesley Publishing Company, 1981).

(define pi 3.14159265358979323846264338327950288419716939937510)
           ; according to the _CRC standard mathematical tables_

(define expected-value (- 1 (/ 6 pi pi)))

;; Given a non-negative integer TEST-COUNT, the INACCURACY procedure
;; determines the absolute value of the difference between the reduction
;; frequency as determined by REDUCING-RATE and the expected value
;; predicted by Dirichlet.

(define inaccuracy
  (lambda (test-count)
    ; Complete this procedure.
    ))