3.1.5 Argument Reduction

The accuracy of may only be known over a restricted domain. Consider the sine function, which may be approximated by a finite polynomial:

Although there are better ways of approximating the sine function, the McLaurin polynomial above will duly serve our purposes.

The function is reasonably accurate for but it is not accurate for large angles. Large arguments are reduced by exploiting the trigonometric identity

as follows:

.

The usual method for accurately computing the reduced argument employs high precision floating point arithmetic with an accurate representation of . Such an approach would use to compute an accurate argument reduction when computing sine for . Some systems do not compute highly accurate reduced arguments hence it may be difficult to accurately estimate the error of for large x. Conventional argument reduction is discussed in [22, 47].

Another approach is to compute the argument reduction using interval arithmetic, and invoke the provided function for small angles, where the error is known. First, the reduced angle is bounded:

Then, a suitable is chosen and the provided function is invoked with x' as an argument. Since is an accurate function over b, the previous subsection on accurate functions details how is constructed from . An upper bound on is similarly constructed from with . Which is chosen depends on whether a lower or upper bound is desired. Sections  and describes how x' is chosen.

Jeff Tupper

March 1996