Nonstandard Floating Point

IEEE 754 floating-point default arithmetic is "nonstop." Underflows are "gradual." Following is a summary of explanation. See the Numerical Computation Guide for details.

Nonstop means that execution does not halt on occurrences like division by zero, floating-point overflow, or invalid operation exceptions. For example, consider the following, where x is zero and y is positive:

z = y / x;

By default, z is set to the value +Inf, and execution continues. With the -fnonstd option, however, this code causes an exit, such as a core dump.

Here is how gradual underflow works. Suppose you have the following code:

x = 10;
for (i = 0; i < LARGE_NUMBER; i++)
	x = x / 10;

The first time through the loop, x is set to 1; the second time through, to 0.1; the third time through, to 0.01; and so on. Eventually, x reaches the lower limit of the machine's capacity to represent its value. What happens the next time the loop runs?

Let's say that the smallest number characterizable is:

1.234567e-38

The next time the loop runs, the number is modified by "stealing" from the mantissa and "giving" to the exponent:

1.23456e-39

and, subsequently,

1.2345e-40

and so on. This is known as "gradual underflow," which is the default behavior. In nonstandard behavior, none of this "stealing" takes place; typically, x is simply set to zero.