expm1x

expm1x is a mathematical function defined as exp(x) − 1, where exp denotes the exponential function e^x. It is closely related to the commonly used expm1 function in numerical software, and in many libraries expm1x serves as an alias or variant of expm1. The naming may vary by implementation, with some contexts explicitly calling the operation expm1x to emphasize a version suitable for vectorization, extended precision, or complex inputs, while others simply use expm1.

For real inputs, expm1x maps the real line onto the interval (-1, ∞), with expm1x(0) = 0. Its derivative with respect to x is exp(x), which can also be written as 1 + expm1x since exp(x) = 1 + (e^x − 1). The function has the standard series expansion expm1x = x + x^2/2! + x^3/3! + …, which is especially accurate for small |x| and underpins many stable numerical implementations.

Numerical considerations are central to expm1x. For small x, computing e^x − 1 directly can suffer from catastrophic cancellation, so stable implementations use the series or other specialized techniques to maintain precision. For large positive x, e^x grows rapidly; while expm1x ≈ e^x, practical computations may still be limited by overflow in e^x, so users rely on library routines designed to handle such cases. For large negative x, expm1x approaches −1 with high accuracy.

Although expm1x is not universally standardized as a separate function name, it commonly appears as an alias or variant of expm1 in numerical libraries and documentation. See also exp, expm1, and log1p.