Mixture of Lebesque and Riemann integral

Here are 3 types of integration, the first is the usual Riemann integration where the domain is split evenly into rectangles and the second is Lebesque where the range is split into even intervals. The bottom one in this picture is the one I thought of where you use whichever results in fewer rectangles for a given delta size over an interval. I was thinking for functions that are very expensive on a computer to evaluate it would speed up the evaluation of the integral.