Ulp

ulp stands for unit in the last place. In floating-point arithmetic, ulp(x) is the distance between x and the next representable floating-point number greater than x, i.e., the gap to nextafter(x, +inf). This quantity measures the spacing of representable numbers near x and is always nonnegative. It can be used to assess the potential rounding error when x is subjected to arithmetic operations.

In IEEE 754 binary formats, the spacing depends on the magnitude of x. For a format with p-bit significand, the typical gap between consecutive normal numbers in the same binade [2^k, 2^{k+1}) is about 2^{k - p + 1}. As x approaches zero, the presence of subnormal numbers reduces gaps, and ulp(0) equals the distance to the smallest positive subnormal value. For infinities and NaN, ulp is not defined in the same way.

Relation to error analysis: ulp is related to machine epsilon, which is ulp(1). It provides a local measure of rounding error and is used to bound relative errors, compare floating-point values, and guide numerical algorithms that require precision control. Because ulp varies with x, absolute tolerances must scale with ulp when working across a wide range of magnitudes.

Implementation and usage: Some languages provide a dedicated ulp function, such as Math.ulp in Java, to return ulp(x). Other environments rely on the concept of machine epsilon or implement ulp via nextafter operations. Understanding ulp helps in debugging numerical issues and in designing robust floating-point computations.