Numerically Stable Solutions for Cubic Equations: Eliminating Catastrophic Cancellation in Cardano's Formula

This paper presents numerically stable reformulations for solving cubic equations, with particular emphasis on eliminating catastrophic cancellation inherent in Cardano's formula. Two explicit closed-form solutions are developed for the cases b² = 3ac and b² < 3ac, including a novel formulation based on auxiliary variables that ensures high numerical stability near discriminant degeneracy. The proposed methods avoid subtraction of nearly equal quantities and are shown to significantly improve numerical accuracy compared to classical formulations of Cardano's method. Illustrative examples demonstrate the effectiveness and robustness of the proposed formulas, establishing them as reliable alternatives in scientific computing.