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3.1.5 Argument Reduction

The accuracy of tex2html_wrap_inline33805 may only be known over a restricted domain. Consider the sine function, which may be approximated by a finite polynomial:

math11042

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

The tex2html_wrap_inline33937 function is reasonably accurate for tex2html_wrap_inline33939 but it is not accurate for large angles. Large arguments are reduced by exploiting the trigonometric identity

math11056

as follows:

tex2html_wrap_inline33941 .

The usual method for accurately computing the reduced argument employs high precision floating point arithmetic with an accurate representation of tex2html_wrap_inline33943 . Such an approach would use tex2html_wrap_inline33945 to compute an accurate argument reduction when computing sine for tex2html_wrap_inline32373 . Some systems do not compute highly accurate reduced arguments hence it may be difficult to accurately estimate the error of tex2html_wrap_inline33949 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 tex2html_wrap_inline33937 function for small angles, where the error is known. First, the reduced angle is bounded:

math11073

Then, a suitable tex2html_wrap_inline33959 is chosen and the provided tex2html_wrap_inline33937 function is invoked with x' as an argument. Since tex2html_wrap_inline33937 is an accurate function over b, the previous subsection on accurate functions details how tex2html_wrap_inline33969 is constructed from tex2html_wrap_inline33971 . An upper bound on tex2html_wrap_inline32437 is similarly constructed from tex2html_wrap_inline33971 with tex2html_wrap_inline33959 . Which tex2html_wrap_inline33959 is chosen depends on whether a lower or upper bound is desired. Sections gif and gif describes how x' is chosen.


next up previous notation contents
Next: 3.1.6 Basic Methods Up: 3.1 Floating Point Previous: 3.1.4 Accurate Functions
Jeff TupperMarch 1996