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

This paper introduces, formalizes, and analyzes the KB-Raiz Optimized Formula, a novel deterministic, closed-form approximation method for calculating square roots. The method is based on the principle of structured interpolation over perfect squares, enhanced by an empirically derived correction term to achieve high precision. The complete is presented axiomatic foundation, derive the core formula and its optimized variant, and establish its formal properties. A comprehensive comparative analysis against established iterative methods—specifically the Newton-Raphson (Heron's) method and linear interpolation techniques—demonstrates that KB-Raiz provides a superior balance between computational complexity, predictability, and accuracy for a single evaluation. The formula achieves an average relative error of 0.0012% using a fixed sequence of only 11 arithmetic operations, making it exceptionally suitable for embedded systems, real-time applications, and compiler optimizations where consistent execution time is critical.