d_lambertw

Close to zero, the result is $x - x^2$ (for $|x| < 10^{-9}$) or a rational approximation. Close to the branch point $-1/e$, it is a polynomial in $\sqrt{x + 1/e}$, where $x + 1/e$ is computed with a correction term for the rounding of $1/e$ (the most negative double in the domain is $-6627126856707895 \cdot 2^{-54}$). Elsewhere, an initial value from a piecewise rational approximation, or for $x > 40$ from the asymptotic expansion in $L_1 = \log x$ and $L_2 = \log \log x$ [CGHJK1996], is refined by one or two steps of Halley's method, rescaled to avoid overflow of $e^w$ near the top of the exponent range.

A test of $10^7$ random inputs showed a maximum relative error below $0.95 \cdot 2^{-52}$ for positive $x$; for negative $x$ the error is larger, up to about $10 \cdot 2^{-52}$ close to $-1/e$. The accuracy depends on the compiler flags and on the system exp and log. For a rigorous enclosure at any precision, use arb_lambertw.

@misc{flint-doc-double-extras-d-lambertw,
  author       = {{The FLINT team}},
  title        = {{d\_lambertw (double\_extras.h)}},
  howpublished = {FLINT: Fast Library for Number Theory, documentation, version 3.7.0-dev},
  year         = {2026},
  url          = {https://flintlib.org/doc/double_extras/d_lambertw.html},
  note         = {Generated 2026-11-02}
}