Filter
Top Products
■ If an invalid input is provided or if the result cannot be
represented exactly, a floating-point exception must be raised.
Careful Rounding
If the result of an IEEE arithmetic operation cannot be represented
exactly in binary format, the number is rounded. IEEE arithmetic
normally rounds results to the nearest value that can be
represented in the chosen data format. The difference between
the exact result and the represented result is the roundoff error.
The IEEE standard requires that users be able to choose to round
in directions other than to the nearest value. For example,
sometimes you might want to know that rounding has not
invalidated a computation. One way to do that would be to force
the rounding direction so that you can be sure your results are
higher (or lower) than the exact answer.
Because it conforms to the IEEE standard, PowerPC Numerics
gives you a means of doing that. Fully developed, this strategy is
called interval arithmetic (Kahan 1980).The following example is
a simple demonstration of the advantages of careful rounding.
Suppose your application performs operations that are mutually
inverse; that is, operations
y=f(x), x=g(y), such that g(f(x))=x.
There are many such operations, such as
y=x
2
, x=Ãy
y=375x, x=yÖ375
Suppose F(x) is the computed value of f(x), and G(y) is the
computed value of
g(y). Because many numbers cannot be
represented exactly in binary, the computed values
F(x) and G(y)
will often differ from f(x) and g(y). Even so, if both functions are
continuous and well behaved, and if you always round
F(x) and
G(y) to the nearest value, you might expect your computer
arithmetic to return
x when it performs the cycle of inverse
operations,
G(F(x))=x.
It is difficult to predict when this relation will hold for computer
numbers. Experience with other computers says it is too much to
expect, but IEEE arithmetic very often returns the correct inverse
value.
The reason for IEEE arithmeticÕs good behaviour with respect to
inverse operations is that it rounds so carefully. Even with all
operations in, say, single precision, it evaluates the expression
3x1/3 to 1.0 exactly; some computers that do not follow the
standard do not evaluate this expression exactly.
Technical Considerations
iMalc Manual
Technical Considerations
33