A New Approximation for the Perimeter of an Ellipse

We consider the problem of approximating the perimeter of an ellipse, for which there is no known finite formula, in the context of high-precision performance. Ellipses are broadly used in many fields, like astronomy, manufacturing, medical imaging, and geophysics. They are applied on large and nanoscales, and while numerical integration can be used to obtain precision measurements, having a finite formula can be used for modeling. We propose an iterative symbolic regression approach, utilizing the pioneering work of Ramanujan’s second approximation introduced in 1914 and a known Padé approximation, leading to good results for both low and high eccentricities. Our proposed model is also compared with a very comprehensive historical collection of different approximations collated by Stanislav Sýkora. Compared with the best-known approximations in this centuries-old mathematical problem, our proposed model performs at both extremities while remaining consistent in mid-range eccentricities, whereas existing models excel only at one extremity.