Trigonometric Tables - On-demand Computation

On-demand Computation

Modern computers and calculators use a variety of techniques to provide trigonometric function values on demand for arbitrary angles (Kantabutra, 1996). One common method, especially on higher-end processors with floating-point units, is to combine a polynomial or rational approximation (such as Chebyshev approximation, best uniform approximation, and Padé approximation, and typically for higher or variable precisions, Taylor and Laurent series) with range reduction and a table lookup — they first look up the closest angle in a small table, and then use the polynomial to compute the correction. Maintaining precision while performing such interpolation is nontrivial, however; and methods like Gal's accurate tables, Cody and Waite reduction, and Payne and Hanek reduction algorithms can be used for this purpose. On simpler devices that lack a hardware multiplier, there is an algorithm called CORDIC (as well as related techniques) that is more efficient, since it uses only shifts and additions. All of these methods are commonly implemented in hardware for performance reasons.

For very high precision calculations, when series-expansion convergence becomes too slow, trigonometric functions can be approximated by the arithmetic-geometric mean, which itself approximates the trigonometric function by the (complex) elliptic integral (Brent, 1976).

Trigonometric functions of angles that are rational multiples of 2π are algebraic numbers, related to roots of unity, and can be computed with a polynomial root-finding algorithm in the complex plane. For example, the cosine and sine of 2π ⋅ 5/37 are the real and imaginary parts, respectively, of a 37th root of unity, corresponding to a root of a degree-37 polynomial x37 − 1. Root-finding algorithms such as Newton's method are much simpler than the arithmetic-geometric mean algorithms above while converging at a similar asymptotic rate; the latter algorithms are required for transcendental trigonometric constants, however.