Fixed Point Approximation via Segmented Taylor Series

The patent divides the domain of the power function into multiple segments, each centered around a convenient expansion point. By segmenting the input range, the system ensures that any input value falls within a segment where Taylor series approximation maintains high accuracy, thus resolving the contradiction between processing speed and accuracy.

The patent dynamically changes the expansion point parameter based on the input value's location within different segments. This allows the Taylor series to be centered at optimal points for each segment, maintaining accuracy across the entire domain while preserving the computational efficiency of the series approximation method.

The patent applies partial action by using Taylor series approximation with a limited number of terms rather than the full iterative Newton-Raphson method. This provides sufficient accuracy for the application while significantly reducing the processing power requirements and device complexity.

The patent uses computationally inexpensive Taylor series calculations that can be performed with simple arithmetic operations, replacing the more complex and resource-intensive Newton-Raphson iterations. This approach achieves adequate accuracy with much lower computational cost.

The patent segments the large lookup table into multiple smaller tables, each covering a specific segment of the domain. This segmentation reduces the memory footprint of each individual table while maintaining comprehensive coverage of the entire domain through the segmented structure.

The patent introduces an additional dimension by organizing the lookup data into a hierarchical structure with multiple levels. This dimensional reorganization allows efficient access to pre-computed values while reducing the total memory required compared to a single flat table.