Computing new covid-19 cases after correction for daily variation in testing capacity

Here’s how to interpret the variation of new official daily infected, taking into account the large fluctuation in the daily number of tests from one day to the next and the progressive increase (trend, moving average) of the daily number of tests that has been up to today. Calculating the correlation between the variable X (number of daily tests on day n / daily number of tests on day n-1) and variable Y (number of daily infected persons on day n / number of daily infected persons on day n-1), I deduced that if the daily number of tests increases by y more (from one day to the next, therefore with the number of actual infected people practically equal) then the daily number of infected people detected (confirmed) increases by y ^ ½ times. That is, for example, if, at parity of new real infected (all and not only those detected) I perform twice the tests I detect the root of 2 times more. NOW WE EXTRACT THE FORMULA that will allow us to calculate the variation, expressed as the increase k of the new real daily infected (k times, where k is a real positive number) knowing the increase (y) of the daily number of tests and the increase ( x) of the daily number of infected persons, that is, official. Bear in mind that x and y are also positive real numbers, i.e. they can be less than 1 (decrease).
It is evident that k (y / k) ^ 1/2 = x, then (k ^ (1/2)) (y ^ (1/2)) = x, then (k ^ (1/2) = x / (y ^ (1/2), so we have that k = (x ^ 2) / y.
Let’s take an example: if the official daily infected increase by 0.5 times that is halved and the daily tests increase by 2 times, then THE ACTUAL NEW DAILY CASES are actually increased by (0.5 ^ 2) / 2 times, that is DECREASED by 8 times !!!