When little information is available on the provided functions,
or the provided functions are unsatisfactory, rigorous upper
and lower bounds may be computed by resorting to basic methods [22, 47, 29].
The methods employed will depend on the function to be computed,
but usually a method for computing 
can be adapted
to compute 
or 
.
With a thorough understanding of the method used to compute ,
an efficient, similar method may be used to compute or
. Let us consider the sine function:
,
we may infer that x, 
, and 
are all non-negative. We may now bound 
using
the provided floating point operations, as follows:
and 
is now clear, for 
. Using similar
bounds on for 
,
and
may be evaluated for 
.
A similar implementation of the cosine function for
along with appropriate argument reduction allows ,
, 
, and 
to be evaluated for all finite floating-point numbers x.
Infinite arguments may be handled by table lookup.
The interval based argument reduction presented in the
last subsection will correctly handle infinite arguments without
table lookup.
Even with limited understanding of the method used to compute , both
and may be implemented.
Again, we consider the sine function:
:
would compute the interval
, as shown above, and return l. An implementation of
would return u. The implementation may be extended,
as the first was, to allow and
to be computed for arbitrary arguments.
Although the two methods initially appear to be distinct, the first implementation of
is simply a cleverly optimized version of the second implementation
of . Further optimization is possible.
For example, one may precompute constants so that division operations
may be replaced with multiplication operations:
and
from 
and 
,
using the method of computing from 
as a guide.
Knowledge of 
and 
can help produce good implementations
of and .
| Jeff Tupper | March 1996 |