 Cooley–Tukey, have excellent numerical properties as a consequence of the pairwise summation structure of the algorithms.The upper bound on the relative error for the Cooley–Tukey algorithm is O("ε" log "N"), compared to O("εN"3/2) for the naïve DFT formula, where ε is the machine floating-point relative precision.In fact, the root mean square (rms) errors are much better than these upper bounds, being only O("ε" ) for Cooley–Tukey and O("ε" ) for the naïve DFT (Schatzman, 1996).