Examination of iterative solutions of the Colebrook-White equation: highest accuracy and least CPU time with one single logarithm calculation

Abstract About eight centuries ago the Colebrook-White equation has been developed to describe the friction factor of turbulent flow in pipes in the region from full smooth to full rough surfaces. Because this equation is implicit in the friction factor many solutions have been developed to simplify the calculation while considering a reasonable accuracy. In 2025 Große-Dunker [1] has proposed a one-step iterative solution, that requires the calculation of only two logarithm functions and provides highest accuracy as well as least CPU time. In this paper, for a further reduction of the CPU load, several iterative solutions are developed and examined, that only need one single logarithm calculation. In the first part of this examination these solutions are compared against each other in a chart of CPU time versus accuracy, the three best provide errors of 1.0·10‑4, 1.9·10-6 and 4.1·10‑8. In the second part these three solutions are compared with 34 solutions from the literature, among them 6 solutions that also require only one logarithm calculation, and they prove to be the fastest in the accuracy range of interest. Conclusion This examination proves, that the Colebrook-White equation can be solved very fast and very accurately with iterative approaches, that require besides basic arithmetic operations only one single logarithm calculation. The recommended solutions are based on a simple concept, consisting of an accurate initial starting point and a one-step iteration with a strong convergence method. By setting the order of the convergence method to 2, 3 or 4, the error of the solutions can be tailored to 1.0·10‑4, 1.9·10‑6 and 4.1·10‑8, the related accuracies are 4.0, 5.7 and 7.4 minimal accurate digits (MAD). The comparison with 34 solutions from literature, among them 6 solutions that also require only one single logarithm calculation, shows that they are the fastest in the range of 2.5 to 9.3 MAD.