log1pl

Log1pl is the long double variant of the logarithm of one plus x, written as log(1+x) with long double precision. It is part of the C standard library’s math API, typically declared as long double log1pl(long double x) in math.h as part of C99 and later standards. The function is designed to compute the natural logarithm of 1+x with high accuracy when x is near zero, avoiding cancellation that can occur with a naïve computation of log(1+x).

Domain and behavior: log1pl is defined for real x with x > -1, since 1+x must be positive. For x approaching -1 from above, the result tends to negative infinity. If x is less than or equal to -1, the operation is outside the real domain and the function may raise a domain error, return NaN, or produce -inf depending on the library implementation.

Numerical approach: implementations use careful argument handling, often employing range reduction and a polynomial or rational approximation to maintain accuracy across the real line in long double precision. This reduces relative error for small x and improves numerical stability compared with directly evaluating log(1+x).

Availability: log1pl is widely available in major C libraries that provide long double variants of math functions (such as glibc and musl on Linux, BSD variants, and other platforms). Some environments may provide only log1p (double) or log1pf (float) if a long double variant is not implemented.

See also: log1p, log1pf, log1pl. Applications include numerical analysis, statistical computations, and scientific simulations where stable logarithm calculations near x = 0 are important.