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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.
Next: 3.1.6 Basic Methods
Up: 3.1 Floating Point
Previous: 3.1.4 Accurate Functions