# lambert_w
[](https://crates.io/crates/lambert_w)
[](https://docs.rs/lambert_w/latest/lambert_w/)
[](https://github.com/JSorngard/lambert_w)
[](https://github.com/JSorngard/lambert_w/actions/workflows/rust.yml)
[](https://codecov.io/gh/JSorngard/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](https://en.wikipedia.org/wiki/Lambert_W_function)
with the method of Toshio Fukushima \[[1](#references)\].
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](https://en.wikipedia.org/wiki/Omega_constant) with the
principal branch of the Lambert W function:
```rust
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:
```rust
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:
```rust
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:
```rust
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`](https://docs.rs/approx/latest/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:
```rust
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`](https://crates.io/crates/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](https://doi.org/10.13140/RG.2.2.30264.37128).
November 2020.
[⬆️ Back to top](#lambert_w).
<br>
### License
<sup>
Licensed under either of <a href="LICENSE-APACHE.txt">Apache License, Version
2.0</a> or <a href="LICENSE-MIT.txt">MIT license</a> at your option.
</sup>
<br>
<sub>
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.
</sub>