The KB-Raiz Formula: A Novel Closed-Form Approximation for Square Root Computation with Integrated Optimized Correction

This work presents a closed-form approximation for the computation of the square root of positive real numbers, referred to as the KB-Raiz formula. The method is based on structured Hermite interpolation between consecutive perfect squares, exploiting the analytical knowledge of the function √x and its derivative at these natural nodal points. The resulting formulation leads to a non-iterative expression for approximating √x whose arithmetic complexity is fixed and independent of the input value. In addition to the fundamental approximation obtained through cubic interpolation, an optimized correction term is introduced to model the dominant behavior of the residual interpolation error. This additional term significantly reduces the mean relative error of the approximation across a broad numerical domain while preserving a low computational cost. The proposed formulation is analyzed from a theoretical perspective and compared with classical strategies for square-root computation, including iterative methods and direct linear approximations. The results indicate that the KB-Raiz formula provides a deterministic approximation with good practical accuracy and a fixed number of arithmetic operations, making it potentially useful in computational contexts where execution-time predictability and algorithmic simplicity are important requirements.