Crate ellip 

Ellip - Elliptic integrals for Rust

Ellip is a pure-Rust implementation of elliptic integrals. This means there is no dependence on C++ libraries. Ellip also provides less common functions like Bulirsch’s cel and el. Some applications of elliptic integrals include computing the lengths of plane curves, magnetic field from magnets of various shapes (e.g., cylindrical), astrophysics, and string theory.

Use Ellip-Rayon to parallelize and improve performance for large inputs. Ellip is also available for Python via EllipPy.

Example

Calculating the perimeter of an ellipse.

use ellip::*;

fn ellipse_perimeter(a: f64, b: f64) -> Result<f64, StrErr> {
    Ok(8.0 * elliprg(0.0, a * a, b * b)?)
}

// Example: ellipse with semi-major axis 5, semi-minor axis 3
println!("{}", ellipse_perimeter(5.0, 3.0).unwrap()); // 25.526998863398124

Features

• legendre — Legendre’s integrals
  • Complete integrals: the first, second, and third kinds (ellipk, ellipe, ellippi), a combined routine (ellipke), and Legendre’s type (ellipd)
  • Incomplete integrals: the first, second, and third kinds (ellipf, ellipeinc, ellippiinc) and Legendre’s type (ellipdinc)
• bulirsch — Bulirsch’s integrals
  • Complete integrals: the general complete integral (cel), and the first, second, and third kinds (cel1, cel2, cel3)
  • Incomplete integrals: the first, second, and third kinds (el1, el2, el3)
• carlson — Carlson’s symmetric integrals: the first, second, and third kinds (elliprf, elliprg, elliprj), and the degenerate integrals (elliprc, elliprd)
• misc — Miscellaneous functions: jacobi_zeta and heuman_lambda

Testing

The full accuracy report, test datasets, and test generation scripts are available in the tests directory. The test and benchmark summary is available in TESTING.md. See how to reproduce the results in REPRODUCIBILITY.md.

Acknowledgment

Ellip is derived from multiple mathematic libraries. We thank the opensource contributors for making mathematic libraries free for all. Following are the main original works used in the development of Ellip. Detailed credits are available in the source code.

• SciPy
• Cephes Math Library
• Boost Math Library
• Russell Lab

References for original implementations are:

• NIST Digital Library, Chapter 19: Elliptic Integrals (Carlson, 2025).
• Numerical calculation of elliptic integrals and elliptic functions I (Bulirsch, 1965), II (Bulirsch, 1965), and III (Bulirsch, 1969).
• NIST Digital Library, Chapter 22 Jacobian Elliptic Functions (Reinhardt and Walker, 2025).

Unicode-style mathematical notation are created using Diagon.

Citation

The paper describing Ellip is published in the Journal of Open Source Software. If Ellip is helpful to your work, please consider citing it:

Pornsiriprasert, S., (2026). Ellip: An Elliptic Integral Library for Rust. Journal of Open Source Software, 11(118), 9386, https://doi.org/10.21105/joss.09386

Bibtex format:

@article{Pornsiriprasert2026,
    doi = {10.21105/joss.09386},
   url = {https://doi.org/10.21105/joss.09386},
   year = {2026},
   publisher = {The Open Journal},
   volume = {11}, number = {118}, pages = {9386},
   author = {Pornsiriprasert, Sira},
   title = {Ellip: An Elliptic Integral Library for Rust},
   journal = {Journal of Open Source Software} }

Re-exports

pub use legendre::ellipd;

pub use legendre::ellipe;

pub use legendre::ellipk;

pub use legendre::ellipke;

pub use legendre::ellippi;

pub use legendre::ellipdinc;

pub use legendre::ellipeinc;

pub use legendre::ellipf;

pub use legendre::ellippiinc;

pub use legendre::ellippiinc_bulirsch;

pub use bulirsch::cel;

pub use bulirsch::cel1;

pub use bulirsch::cel2;

pub use bulirsch::cel3;

pub use bulirsch::el1;

pub use bulirsch::el2;

pub use bulirsch::el3;

pub use bulirsch::el3;

pub use carlson::elliprc;

pub use carlson::elliprd;

pub use carlson::elliprf;

pub use carlson::elliprg;

pub use carlson::elliprj;

pub use misc::heuman_lambda;

pub use misc::jacobi_zeta;

Modules

bulirsch

Elliptic integral functions in Bulirsch’s form.

carlson

Elliptic integral functions in Carlson’s form.

legendre

Elliptic integral functions in Legendre’s form.

misc

Functions related to elliptic integrals

util

Utility functions like assert_close.

Type Aliases

StrErr

Static error str