Abstract—Computing n! efficiently and accurately is notoriously difficult. Many have proposed approximations, varying in accuracy and in computational complexity. Interesting and useful approxi¬mations are both accurate and computationally inexpensive, and, if possible, exact up to machine-precision floating-point numbers. In this paper, we exploit an observation on a previous approximation by Hodgman to obtain a new class of correction terms using simple, but optimal given their degree, rational functions. We show that the proposed approximations are more accurate than some of the best-known approximations while remaining computationally inexpensive.

Index Terms— Approximation, asymptotic error, factorial, rational function.

Steven Pigeon is with Department of Mathematics, Computer Science and Engineering, Université du Québec à Rimouski, Canada.

Cite:Steven Pigeon, "Simple but High-Accuracy Approximations for n!," International Journal of Applied Physics and Mathematics vol. 11, no. 4, pp. 61-70, 2021.

Copyright © 2021 by the authors. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).