How to find the function f that satisfies this

; The original problem:

; I want to find out the fn that satisfies this test for any x > 0 and y > 0, supposing x are generally (but not guaranteed) aprox. y.

(define ok (lambda (fn x y)
             (let ((result1 (fn x y))
                   (result2 (/ 1 (fn (/ 1 x)
                                     (/ 1 y)))))
             (and (= result1 result2)
                  (< (min x y) result1)
                  (> (max x y) result1)))))

; So I created this question: http://math.stackexchange.com/questions/985506/how-to-find-the-function-f-that-satisfies-fx-y-fx-1-y-1-1-a

; Then I defined:

(define fn-solution (lambda (x y)
                       (sqrt (* x y))))

; But:

(ok fn-solution 5 8) => #f

; for example. Why did not work? Because sqrt is not precise, it is a irrational number.

; So, if x and y are exact numbers, then this solution works:

(define fn-final-solution
  (lambda (x y)
    (let ((use-inverse (< (min x y)
                          (min (/ 1 x) (/ 1 y)))))
      (if (not use-inverse)
          (inexact->exact (sqrt (* x y)))
          (/ 1 (inexact->exact (sqrt (* (/ 1 x)
                                        (/ 1 y)))))))))

; The final test:

(define nice-random
  (lambda ()
    (inexact->exact (/ (random) (random)))))

(let next ((x (nice-random))
           (y (nice-random))
           (counter 0))
  (or (> counter 100000)
      (let ((result (ok fn-final-solution x y)))
        (and result
             (next (nice-random)
                   (nice-random)
                   (+ counter 1))))))

; Nice random is just to pick a random number sometimes less than 0 and sometimes bigger. Also it is exact number.

This test returns #t