floorlog2x

Floorlog2x, commonly written as floor(log2(x)), is the greatest integer less than or equal to the base-2 logarithm of x. It is defined for positive x (x > 0). By definition it satisfies 2^k <= x < 2^(k+1), where k = floor(log2(x)). Thus floorlog2x returns the exponent of the highest power of two not exceeding x.

For integer inputs, floorlog2x also equals the index of the most-significant 1 bit in the binary representation of x, counting from zero. Consequently, the number of bits required to represent x in binary is floorlog2x + 1. If 0 < x < 1, floorlog2x is negative (for example floorlog2(0.5) = -1).

Computing floorlog2x can be done in several ways. A straightforward approach uses a logarithm function and truncation, though this can introduce rounding errors with floating-point arithmetic. More robust methods rely on integer operations: finding the position of the most-significant set bit, or using count-leading-zeros facilities available in many programming environments. These approaches typically run in constant time on fixed-width integers.

Common applications include algorithm analysis and data structure design, such as determining the number of iterations in binary search, the depth of a binary tree, or the maximum block count in divide-and-conquer algorithms. It also appears in problems involving binary representations, memory allocation, and coding theory, where powers of two define natural boundaries.