Hacker's Delight

3-2. Rounding Up/Down to the Next Power of 2

We define two functions that are similar to floor and ceiling, but which are directed roundings to the closest integral power of 2, rather than to the closest integer. Mathematically, they are defined by

The initial letters of the function names are intended to suggest “floor” and “ceiling.” Thus, flp2(x) is the greatest power of 2 that is ≤ x, and clp2(x) is the least power of 2 that is ≥ x. These definitions make sense even when x is not an integer (e.g., flp2(0.1) = 0.0625). The functions satisfy several relations analogous to those involving floor and ceiling, such as those shown below, where ...