The floor, computed two ways

The floor, computed two ways

The exact bound log₂(n!) summed term by term, Stirling's closed form laid over it, and n log₂ n above both for comparison. The exact sum and the approximation agree to better than one part in a million across this range. The gap to n log₂ n is n log₂ e, about 1.44n, which is why an algorithm at exactly n log₂ n comparisons is not optimal but is within a factor that shrinks towards 1.

Drawn at 660 × 400, wide on the page. Everything above is what floor-curve returns with no arguments; the caption is the generator's own, computed from the numbers in the drawing rather than written beside it.

6 essays call floor-curve. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about — which is what optcheck and figfill exist to catch.