3.1.4 Accurate Functions

Consider the n-ary function , along with the functions and which overestimate the error of , an accurate model of . The function overestimates the amount by which exceeds while overestimates the amount by which exceeds :

A simple implementation of would proceed as follows:

a simple implementation of would proceed as follows:

In many cases, and would be computed concurrently with and would use partial results computed during the computation of . If necessary, table lookup may be used to handle infinite arguments.

As an example, consider a model , of , which guarantees:

which is equivalent to stating that no floating point number is closer to than . The functions

correctly overestimate the error of , where and give the floating-point number immediately preceding and succeeding x, respectively:

Using the preceding error overestimates, a simple implementation would proceed as follows:

Jeff Tupper

March 1996