Third, missing tile count is irrelevant. It is effectively a measure of area and you said it yourself in the second paragraph of reply #20 (paraphrased) that area is not a good measure of effectiveness.
It is not a missing tile count. It is an estimate of the number of inputs for which the circular approximation gives the wrong answer. Using a ceiling function, any number of tiles from 4219 to 4446 should give a sensor range of 38. The circular approximation assigns an incorrect sensor range to 36% of the tile counts in this range. Using the ceiling function, any number of tiles from 19 to 36 should give a sensor range of 3; the circular approximation gives an incorrect sensor range for 44% of these values. Using a ceiling function, any number of tiles from 37 to 60 should give a sensor range of 4; the circular approximation gives an incorrect sensor range for 42% of these values. Using a ceiling function, any number of tiles from 61 to 90 should give a sensor range of 5; the circular approximation gives an incorrect sensor range for 40% of these values. And so on.
I am not evaluating 'missing tile count.' I am evaluating how effective the circular approximation is at providing the right answer. For ~26% of the possible input values on the range [1, 720], the circular approximation gives the wrong sensor range. This is a bad estimate; the circular approximation provides the wrong answer for more than a quarter of the input tile areas.
First, you have to use ceil, floor, or round somewhere for any and all functions, including yours, becease, as you said yourself, we're talking about discreet units. So don't blame the ceil function, it was chosen to give a little extra range to players for gameplay reasons.
Where did I blame the ceiling function? I said that when using a ceiling function to round the numbers, the circular approximation provides the wrong sensor range for ~26% of tile areas from 1 to 720 tiles. It's about the same error rate as with the floor function (which provides the wrong sensor range for ~24% of the tile areas in the same range) and standard rounding (which provides the wrong sensor range for ~25% of the tile areas in the same range).
I am criticizing the circular approximation, not blaming the error on the ceiling function. I used the same function to round the numbers using the R = -0.5 + sqrt(9 + 12A) formula when determining how many of the values produced by the circular approximation were wrong because a simple comparison of the resulting ranges is the easiest way to get a count of the number of incorrect ranges.
I agree with you that area does not give players much useful info. But again I never said players need to know about area, I'm just saying area is a reasonable method for implementing diminishing returns.
You want sensor components to give bonuses in terms of area so as to implement a form of diminishing returns, and you don't feel that players need to know about area, and you agree that area revealed is not a particularly helpful statistic for the player to know. You are proposing, therefore, that a sensor component's statistics should list its size, its maintenance cost, and its production cost, and perhaps a statistic which is not particularly helpful to the player or no statistics related to its bonus at all. How then should the player decide which sensor component to use? Trial and error? Trial and error is not a fun or interesting way to handle ship design optimization.