Name of the highest power of 2 smaller than or equal to a given number

Solution 1:

In the context of data structures - specifically, the van Emde Boas layout - I've heard this referred to as the hyperfloor of $x$. See “Cache-Oblivious B-Trees (Wayback Machine) by Bender, Demaine, and Farach-Colton for details - it defines the hyperfloor of $x$, denoted $\lfloor \lfloor x \rfloor \rfloor$, to be $2^{\lfloor \log_2 x \rfloor}$.

Hope this helps!

Solution 2:

If $x$ is an integer, then $n+1$ is the bit length of $x$.

Solution 3:

You could call it $2^{\lfloor \log_2(x) \rfloor}$.

Solution 4:

Moved from comment

I suggest derivations from octave or binade. These words are typically used to describe ranges spanning powers of two rather than a specific value within them, but because a power-of-two uniquely defines an octave/binade (it is its lowest value; in other words, it is a base value, or a floor), you could use the same word for both the interval and this defining value.

For instance, you could name it the binadic floor. I favour this expression because the concept of binades is clear once formulated, and the concept of floor is unambiguous and generally well-understood. The rarity of "binade" means that "binadic floor" is sufficiently unusual as to not be confused with the regular floor, and "binadic floor" rolls quite well off the tongue.

Another possibility is octaval base/root/floor, but "octave", "base" and "root" all have pre-existing connotations, and when spoken, octaval floor doesn't sound right because the -al and fl- interact poorly, forcing a break.

Let us define the binade of a positive integer $N$ as the set of integers $[2^n, 2^{n+1})$ that contains $N$, and the binadic floor of an integer $N$ as the lower bound of $N$'s binade...

Solution 5:

I would just describe it as the "largest power of two not exceeding $x$". See A053644 in OEIS for other names and information.