An Exponential Correction to Ramanujan’s Second Formula for Ellipse Perimeter Computation

The exact perimeter of an ellipse involves the complete elliptic integral of the second kind, which lacks a closed-form expression in elementary functions. As a result, analytical approximations have been proposed for applications requiring fast and accurate evaluation of elliptical geometries. In this study, we present a new ultra-accurate and compact closed-form approximation for the ellipse perimeter based on an exponential correction applied to Ramanujan’s second formula. The proposed expression preserves simplicity—using only three exponential functions and six constants—while achieving a maximum relative error of approximately 0.57 ppm observed over the tested grids covering the full eccentricity range. This represents a significant accuracy improvement over classical and modern approximations while maintaining a single-line analytical form with low computational cost. Due to its robustness, quasi-exact behavior at both circular and highly eccentric limits, and its suitability for numerical algorithms and embedded implementations, the proposed approximation is particularly useful in engineering computations involving elliptical boundaries.