Specifying floating-point functions

Consider, for example, the natural logarithm. For positive, finite arguments, the library function Log matches the mathematical function log to within platform-dependent rounding errors. Its special cases are:

1. Log returns a NaN and raises invalid for , because is undefined there.
   
2. Log returns and raises the divide-by-zero flag (to indicate the finite singularity) at the argument because . By convention, it behaves the same at .
   
3. Log returns at the argument , because .
   

It is by convention, and in the spirit of at zero, that functions like Log raise the divide-by-zero flag at finite singularities. Whenever a function raises the invalid flag and the result is of floating-point type, the function returns a NaN; when the result is of integral type, the function returns a platform-dependent integral value.

Note that despite its limits at zero and , Log never overflows. The value of Log at the maximum double value is under 710. Log doesn't underflow, either, because of the relative sparsity of floating-point numbers near 1. The fact that is nearly x for small x says that the value of Log at the double number just larger than 1 is approximately .