To illustrate the savings of an FFT, consider the count of complex multiplications and additions for N=4096 data points. Evaluating the DFT's sums directly involves "N"2 complex multiplications and "N"("N" − 1) complex additions, of which formula_12 operations can be saved by eliminating trivial operations such as multiplications by 1, leaving about 30 million operations. In contrast, the radix-2 Cooley–Tukey algorithm, for "N" a power of 2, can compute the same result with only ("N"/2)log2("N") complex multiplications (again, ignoring simplifications of multiplications by 1 and similar) and "N" log2("N") complex additions, in total about 30,000 operations a thousand times less than with direct evaluation. In practice, actual performance on modern computers is usually dominated by factors other than the speed of arithmetic operations and the analysis is a complicated subject (for example, see Frigo & Johnson, 2005), but the overall improvement from formula_1 to formula_2 remains.