lambert_w

This crate provides fast and accurate evaluation of the real valued parts of the principal and secondary branches of the Lambert W function with the method of Toshio Fukushima [1]. It also provides a slower iterative evaluation method for all branches on the complex plane.

Fukushima's method does not allocate, recurse, or iterate. It works by dividing the function's domain into subdomains. On each one, it uses a simple transformation of the input inserted into a rational function to approximate the true value.

The implementation uses conditional switches on the input value to select the appropriate subdomain, followed by either a square root (and possibly a division) or a logarithm. Then it performs a series of additions and multiplications by constants from a look-up table, and finishes the calculation with a division.

This crate provides two approximations of each branch, one with 50 bits of accuracy (implemented on 64-bit floats) and one with 24 bits (implemented on 32- and 64-bit floats). The one with 50 bits of accuracy uses higher degree polynomials in the rational functions compared to the one with only 24 bits, and thus more of the multiplications and additions by constants.

This crate can evaluate the approximation with 24 bits of accuracy on 32-bit floats, even though it is defined on 64-bit floats in Fukushima's paper. This may result in a reduction in the accuracy to less than 24 bits, but this reduction has not been quantified by the author of this crate.

This crate is no_std compatible, but can optionally depend on the standard library through features for a potential performance gain.

Examples

Compute the value of the omega constant with the principal branch of the Lambert W function:

use lambert_w::lambert_w0;
use approx::assert_abs_diff_eq;

let Ω = lambert_w0(1.0);

assert_abs_diff_eq!(Ω, 0.5671432904097839);

Evaluate the secondary branch of the Lambert W function at -ln(2)/2:

use lambert_w::lambert_wm1;
use approx::assert_abs_diff_eq;

let mln4 = lambert_wm1(-f64::ln(2.0) / 2.0);

assert_abs_diff_eq!(mln4, -f64::ln(4.0));

Do it on 32-bit floats:

use lambert_w::{lambert_w0f, lambert_wm1f};
use approx::assert_abs_diff_eq;

let Ω = lambert_w0f(1.0);
let mln4 = lambert_wm1f(-f32::ln(2.0) / 2.0);

assert_abs_diff_eq!(Ω, 0.56714329);
assert_abs_diff_eq!(mln4, -f32::ln(4.0));

The implementation can handle extreme inputs just as well:

use lambert_w::{lambert_w0, lambert_wm1};
use approx::assert_relative_eq;

let small = lambert_wm1(-f64::MIN_POSITIVE);
let big = lambert_w0(f64::MAX);

assert_relative_eq!(small, -714.9686572379665);
assert_relative_eq!(
    big,
    703.2270331047702,
    // Since the approximation used in this
    // example is accurate to 50 bits
    // it will sometimes have an
    // error larger than epsilon.
    max_relative = 1.5 * f64::EPSILON
);

The macros in the examples above are from the approx crate, and are used in the documentation examples of this crate. The assertion passes if the two supplied values are the same to within floating point epsilon, or within an optional absolute or relative difference.

Arbitrary branches in the complex plane

Functions are provided that can evaluate any arbitrary branch at any arbitrary complex input:

use lambert_w::{lambert_w, lambert_wf};

// W_10(2.5 - 3i)
let w10 = lambert_w(10, 2.5, -3.0);

assert_eq!(w10, (-2.738728537647321, 60.33964127931528));

// Same but 32-bit
let w10f = lambert_wf(10, 2.5, -3.0);

assert_eq!(w10f, (-2.7387285, 60.33964));

These functions use Halley's method to iteratively compute a solution. While this method is more general than the other provided functions, it can be up to two orders of magnitude slower than them for comparable inputs.

Features

One of the below features must be enabled:

libm (enabled by default): if the std feature is disabled, this feature uses the libm crate to compute square roots and logarithms during function evaluation instead of the standard library.

std: use the standard library to compute square roots and logarithms for a potential performance gain. When this feature is disabled the crate is no_std compatible.

References

[1]: Toshio Fukushima. Precise and fast computation of Lambert W function by piecewise minimax rational function approximation with variable transformation. DOI: 10.13140/RG.2.2.30264.37128. November 2020.

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License

^{Licensed under either of Apache License, Version 2.0 or MIT license at your option.}
_{Unless you explicitly state otherwise, any contribution intentionally submitted for inclusion in the work by you, as defined in the Apache-2.0 license, shall be dual licensed as above, without any additional terms or conditions.}