floorlog2k

floorlog2k refers to the floor of the binary logarithm of k, written as ⌊log2(k)⌋. It is the greatest integer n such that 2^n ≤ k. This definition applies to positive k; for real numbers k > 0 it is the integer part of log base 2 of k, and for integer k ≥ 1 it identifies the position of the most significant bit in the binary representation.

For positive integers, floorlog2k also provides a convenient characterization: if k lies in the interval [2^n, 2^(n+1) − 1], then floorlog2k = n. Consequently, the binary length of k (the number of bits needed to represent k in binary) is floorlog2k + 1.

Examples help illustrate the idea: floorlog2(1) = 0, floorlog2(2) = 1, floorlog2(3) = 1, and floorlog2(4) = 2. Powers of two yield exact exponents, while numbers between powers of two share the same floorlog2 value as the lower power.

In computational contexts, floorlog2k is often used to determine the size class of a value, perform bit-length calculations, or locate the most significant set bit. For integer k, floorlog2k can be computed efficiently as the position of the most significant 1 bit, for example via bit-manipulation techniques such as counting leading zeros: floorlog2k = (bitwidth − 1) − clz(k). Some languages provide direct intrinsic operations; others implement it with loops or lookup methods.

Related concepts include ceil(log2(k)) and log2(k) itself, as well as bit-length, most significant bit, and various algorithms that depend on logarithmic partitioning or data structure sizing. floorlog2k is a fundamental tool in analysis of algorithms and low-level numeric computations.