Log1p

Log1p is a mathematical function that returns the natural logarithm of 1 plus its argument, log1p(x) = log(1+x). It provides higher numerical stability than a direct evaluation of log(1+x) for values of x near zero, where cancellation and loss of precision can occur in finite-precision arithmetic.

Domain and behavior: The function is defined for x > -1; at x = 0, log1p(0) = 0. As x approaches -1 from above, log1p(x) tends to negative infinity. For large x, log1p(x) behaves like log(1+x), but is computed in a numerically stable manner in software libraries to avoid precision loss.

Computational aspects: Implementations use special algorithms, such as series expansions around 0, range reduction, and rational approximations to evaluate log(1+x) without catastrophic cancellation. Most libraries also provide variants for different floating types, such as log1pf and log1pl for float and long double. The result is the natural logarithm; if a different base is required, it must be converted from the natural logarithm.

Language support and usage: log1p is part of the C standard library (log1p in <math.h> with several floating-point variants) and is available in many programming languages, including Python (math.log1p), Java (Math.log1p), MATLAB, R, and Ruby. It is commonly used in statistical and numerical computations to compute log(1+x) with improved precision, such as in log-likelihood calculations and updates that involve small fractional changes.