Efficient n-th Root Computation on Microcontrollers Employing Magic Constants and Modified Newton and Householder Methods

With the growing number of applications in embedded systems—such as IoT modules, smart sensors, and wearable devices—there is an increasing demand for fast and accurate computations on resource-constrained platforms. In this paper, we present a new method for computing n-th roots in floating-point arithmetic based on an initial estimate generated by a “magic constant,” followed by one or two iterations of a modified Newton–Raphson or Householder algorithm. For cubic and quartic roots, we provide C implementations operating in single-precision floating-point format. The proposed algorithms are evaluated in terms of maximum relative error and execution time on selected microcontrollers. They exhibit high accuracy and noticeably reduced computation time. For example, our methods for computing cubic roots outperform the standard library function cbrtf() in both speed and precision. The results may be useful in a variety of fields, including biomedical and biophysical applications, statistical analysis, and real-time image and signal processing.