3.2.22 Common Binary Functions

Unary functions have been discussed fully, so we now turn our attention to binary functions. There are several ways of extending the methods presented to handle binary functions. The timely evaluation of arbitrary binary functions is difficult, so we will first list the binary functions that interest us. These functions are: x+y, x-y, , , , , , and ; we may later refer to these as the common binary functions. We consider the exponential function, , for positive bases only: . The function gives the angle from the origin to the point (x,y):

There is a unique angle satisfying the above equation, unless . The function is defined when there is a unique angle; .

The functions of interest may be rewritten, as follows:

The addition operator is the sole remaining binary operator. Interval evaluation using the above rules for the binary operators will not produce optimal results. However, a cursory implementation of an interval arithmetic may use the above rules. It is reasonable to rewrite division, since the result will be nearly optimal, with a simpler implemention. Rewriting may expose salient features to a symbolic optimizer.

Argument reduction may be performed. For example, we may consider multiplication for positive multiplicands only, by exploiting the following identity:

The interaction between the underlying number system and any argument reduction, or rewriting, should be carefully considered. Careless symbolic manipulation may produce a form which needlessly exacts horrendous round-off during evaluation, and subsequently cause sub-optimal intervals to be returned.

Jeff Tupper

March 1996