Range reduction based on Pythagorean triples for trigonometric function evaluation

Abstract

Software evaluation of elementary functions usually requires three steps: a range reduction, a polynomial evaluation, and a reconstruction step. These evaluation schemes are designed to give the best performance for a given accuracy, which requires a fine control of errors. One of the main issues is to minimize the number of sources of error and/or their influence on the final result. The work presented in this article addresses this problem as it removes one source of error for the evaluation of trigonometric functions. We propose a method that eliminates rounding errors from tabulated values used in the second range reduction for the sine and cosine evaluation. When targeting correct rounding, we show that such tables are smaller, and make the reconstruction step cheaper than existing method. This approach relies on Pythagorean triples generators. Finally, we show how to generate tables indexed by up to 10 bits in a reasonable time and with little memory consumption.

Exact Lookup Tables for the Evaluation of Trigonometric and Hyperbolic Functions

“…In this article, we extend a previous work on the trigonometric functions sin and cos presented in [17] to the hyperbolic functions sinh and cosh, and we show that our concept is general enough to be applied to other functions. More precisely, the contributions of this article are the following: 1) A general method that eliminates rounding errors from tabulated values; 2) And a formally justified algorithm to apply this method to trigonometric and hyperbolic functions.…”

“…In 2015, we proposed a method to tabulate error-free values for the sine and cosine functions [17]. It is an improvement over Gal's tables as tabulated approximations for the sine and cosine are stored exactly, so that extended precision computations involving these values shall be less expensive.…”

“…b) Whenever arcsin(b/c) ≈ π/4, one also has arcsin(a/c) ≈ π/4, and both triples may either fall into two different subintervals of the exact lookup table, which can help reduce the value k (since they share a common hypotenuse), or fall into the same subinterval, which can help find a better corrective term. Hence the first step of PPT generation using the Barning-Hall tree is the following: multiplying the matrices in Equation (4) by the root (3, 4, 5) taken as a column vector, one gets the three new PPTs (5,12,13), (15,8,17), and (21,20,29), and their symmetric counterparts (12,5,13), (8,15,17), and (20,21,29). In the following, note that we always consider the "degenerated" PPT (0, 1, 1), because it gives us an exact corrective term for the first table entry, without making the LCM k grow.…”

“…This entails looking for a value k that is rather small, which can be simplified as a search for a small value in a set of least common multiples. It was shown in [17] that a straightforward approach, consisting in computing the set of all possible LCMs, was quickly out of reach with current technology, as soon as tables had more than a few rows.…”

“…To reduce the computational time, the solution proposed in [17] consists in looking for an LCM directly amongst generated values. We call this solution "exhaustive search" in the sequel of this article, as it always gives the smallest LCM.…”