28.12 Exact Floating-Point Constants

One of the frustrations that numerical programmers have suffered with since the dawn of digital computers is the inability to precisely specify numbers in their programs. On the early decimal machines, that was not an issue: you could write a constant 1e-30 and be confident of that exact value being used in floating-point operations. However, when the hardware works in a base other than 10, then human-specified numbers have to be converted to that base, and then converted back again at output time. The two base conversions are rarely exact, and unwanted rounding errors are introduced.

As computers usually represent numbers in a base other than 10, numbers often must be converted to and from different bases, and rounding errors can occur during conversion. This problem is solved in C using hexademical floating-point constants. For example, +0x1.fffffcp-1 is the number that is the IEEE 754 32-bit value closest to, but below, 1.0. The significand is represented as a hexadecimal fraction, and the power of two is written in decimal following the exponent letter p (the traditional exponent letter e is not possible, because it is a hexadecimal digit).

In printf and scanf and related functions, you can use the ‘%a’ and ‘%A’ format specifiers for writing and reading hexadecimal floating-point values. ‘%a’ writes them with lower case letters and ‘%A’ writes them with upper case letters. For instance, this code reproduces our sample number:

printf ("%a\n", 1.0 - pow (2.0, -23));
    -| 0x1.fffffcp-1

The strtod family was similarly extended to recognize numbers in that new format.

If you want to ensure exact data representation for transfer of floating-point numbers between C programs on different computers, then hexadecimal constants are an optimum choice.