A Numerically Stable Reformulation of Cardano's Method for Cubic Equations with b² > 3ac

This paper presents a reformulation of Cardano-type solutions for cubic equations under the condition b² > 3ac, expressed in terms of invariant parameters R and K. The approach unifies algebraic, trigonometric, and complex formulations into a consistent framework. Particular attention is given to numerical behavior in floating-point arithmetic, especially near degenerate cases where classical Cardano's formula suffers from catastrophic cancellation. A detailed forward error analysis based on the IEEE 754 double-precision model is provided. The proposed method is proven to be backward stable under standard floating-point assumptions. Numerical experiments demonstrate that the proposed method achieves relative errors bounded by 10⁻¹⁴ in challenging scenarios where classical Cardano loses up to 10⁻⁴ accuracy or overflows. The improved robustness is attributed to the scale-invariance of the parameters (R, K) and the mitigation of subtractive cancellation in critical expressions.