Constant-Table Power Series Interpolation for Fast Transcendental Functions
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Solution Overview
Problem
Transcendental functions in microprocessors are computationally intensive, consuming a significant percentage of execution time due to the complexity of power series expansions, which require multiple cycles and expensive operations like floating-point divides and fused multiply-adds, leading to high operational latency.
Innovation Solution
The method involves interpolating a primary interval for convergence of power series expansions, selecting interpolation points for a truncated expansion, evaluating function and derivative values at these points, saving them in a table, and using table lookup and computation to efficiently compute results through rotating, shifting, and masking operations, thereby reducing the number of floating-point operations and latency.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If standard power series expansions are used for transcendental functions, then computational accuracy is maintained, but execution time increases significantly due to multiple cycles and expensive operations
Solution Approach 1:
The patent pre-computes power series expansion coefficients and stores them in lookup tables during system initialization or compilation. When a transcendental function needs to be evaluated, the pre-computed coefficients are retrieved from the table rather than calculated on-the-fly, eliminating the need for expensive floating-point operations during execution and significantly reducing computation time while maintaining accuracy.
Solution Approach 2:
The patent creates simplified copies of the complex power series expansion process by storing pre-calculated coefficient values in lookup tables. Instead of executing the full mathematical expansion algorithm during runtime, the system uses these copied values directly, replacing complex computational sequences with simple table lookups and basic arithmetic operations.
2Measurement precision
If full power series expansions are computed, then function evaluation accuracy is preserved, but operational latency increases due to multiple expensive floating-point operations
Solution Approach 1:
The patent extracts the computationally intensive portion of the power series expansion (the coefficient calculations) and separates it from the runtime evaluation process. These extracted coefficients are pre-computed and stored in lookup tables, allowing the runtime evaluation to use only simple arithmetic operations with the pre-fetched coefficients, thereby reducing operational latency while preserving accuracy.
Solution Approach 2:
The patent performs the complex coefficient computation in advance before the actual function evaluation is needed. By pre-computing and storing the power series coefficients in lookup tables during initialization, the system eliminates the need to perform these expensive calculations during runtime, significantly reducing operational latency while maintaining full evaluation accuracy.
3Reliability
If transcendental functions are computed using standard libraries, then correctness is ensured, but performance is degraded due to computational intensity
Solution Approach 1:
The patent pre-computes and stores power series expansion coefficients in lookup tables during system initialization or compilation. When transcendental functions need to be evaluated, the system retrieves these pre-computed coefficients from the tables and performs simple arithmetic operations, ensuring mathematical correctness while achieving significant performance improvements over standard library implementations.
Solution Approach 2:
The patent creates optimized copies of the power series expansion process by storing pre-calculated coefficients in lookup tables. This copying approach replaces the general-purpose but computationally intensive standard library algorithms with specialized, pre-computed data structures that maintain mathematical correctness while dramatically improving execution performance.
Data Source
AI summary
A primary interval for convergence of at least one power series in a transcendental function is interpolated, while selecting a number of one or more interpolation points for a truncated expansion of the at least one power series by a selected order of truncation. A function and at least one derivative of the function of the truncated expansion of the selected order of truncation is evaluated at the one or more interpolation points. Each separate value evaluated for the function and each of the at least one derivative is saved in a table, wherein the table is looked up for efficiently computing a result of the truncated expansion of the at least one power series.