Both IEEE 754 floating-point hardware, and MPFR, support two kinds of
values that you probably didn’t learn about in school. The first is
*infinity*, a special value, that can be either negative or positive,
and which is either smaller than any other value (negative infinity),
or larger than any other value (positive infinity). When such values
are generated, `gawk`

prints them as either ‘`-inf`’ or
‘`+inf`’, respectively. It accepts those strings as data input and
converts them to the proper floating-point values internally.

Infinity values of the same sign compare as equal to each other. Otherwise, operations (addition, subtraction, etc.) involving another number and infinity produce mathematically reasonable results.

The second kind of value is “not a number”, or NaN for
short.^{102} This is a special value that results
from attempting a calculation that has no answer as a real number.
In such a case, programs can either receive a floating-point exception,
or get NaN back as the result. The IEEE 754 standard recommends
that systems return NaN. Some examples:

`sqrt(-1)`

This makes sense in the range of complex numbers, but not in the range of real numbers, so the result is NaN.

`log(-8)`

−8 is out of the domain of

`log()`

, so the result is NaN.

NaN values are strange. In particular, they cannot be compared with other
floating point values; any such comparison, except for “is not equal
to”, returns false. NaN values are so much unequal to other values that
even comparing two identical NaN values with `!=`

returns true!

NaN values can also be signed, although it depends upon the implementation
as to which sign you get for any operation that returns a NaN. For
example, on some systems, `sqrt(-1)`

returns a negative NaN. On
others, it returns a positive NaN.

When such values are generated, `gawk`

prints them as either
‘`-nan`’ or ‘`+nan`’, respectively. Here too, `gawk`

accepts those strings as data input and converts them to the proper
floating-point values internally.

If you want to dive more deeply into this topic, you can find
test programs in C, `awk`

and Python in the directory
`awklib/eg/test-programs` in the `gawk`

distribution.
These programs enable comparison among programming languages as to how
they handle NaN and infinity values.

Thanks to Michael Brennan for this description, which we have paraphrased, and for the examples.