Method of Precision Increase by Averaging with Application to Numerical Differentiation

If several independent algorithms for a computer-calculated quantity exist, then one can expect their results (which differ because of numerical errors) to follow approximately Gaussian distribution. The mean of this distribution, interpreted as the value of the quantity of interest, can be determined with better precision than what is the precision provided by a single algorithm. Often, with lack of enough independent algorithms, one can proceed differently: many practical algorithms introduce a bias using a parameter, e.g. a small but finite number to compute a limit or a large but finite number (cutoff) to approximate infinity. One may vary such parameter of a single algorithm and interpret the resulting numbers as generated by several algorithms. A numerical evidence for the validity of this approach is shown for differentiation.