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One way to maximize the efficiency of doing right-Reimann sum approximations is by ensuring that you do the most amount of computations in the areas where the function would otherwise contain the most amount of errors (where the slope is greatest). Using simple polynomials (x^a), we can explore which values of x are best to compute right-Reimann sum approximations. I’ve only found an equation describing these x-values in implicit terms, where you cannot find where to optimize unless you have already found the best previous values of x, starting from 0.
Using this implicit equation for the function x^2, this program recursively stores answers in an array and uses the bisection method to compute the approximate next result for 1000 results. The program can easily be edited to find more solutions by simply changing the array size, 'MAX'.
The document with more information describing the problem: https://www.overleaf.com/read/gzmppcyqntqj#5e3e43