Quick Start

Start by installing Ellip.

>> cargo add ellip

Let's compute the circumference of an ellipse.

use ellip::*;

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

let ans = ellipse_length(5.0, 3.0).unwrap();
ellip::util::assert_close(ans, 25.526998863398124, 1e-15);

Learn more at doc.rs.

Features

Legendre's complete integrals
- ellipk: Complete elliptic integral of the first kind (K).
- ellipe: Complete elliptic integral of the second kind (E).
- ellippi: Complete elliptic integral of the third kind (Π).
- ellipd: Complete elliptic integral of Legendre's type (D).
Legendre's incomplete integrals
- ellipf: Incomplete elliptic integral of the first kind (F).
- ellipeinc: Incomplete elliptic integral of the second kind (E).
- ellippiinc: Incomplete elliptic integral of the third kind (Π).
- ellipdinc: Incomplete elliptic integral of Legendre's type (D).
Bulirsch's integrals
- cel: General complete elliptic integral in Bulirsch's form.
- cel1: Complete elliptic integral of the first kind in Bulirsch's form.
- cel2: Complete elliptic integral of the second kind in Bulirsch's form.
- el1: Incomplete elliptic integral of the first kind in Bulirsch's form.
- el2: Incomplete elliptic integral of the second kind in Bulirsch's form.
- el3: Incomplete elliptic integral of the third kind in Bulirsch's form.
Carlson's symmetric integrals
- elliprf: Symmetric elliptic integral of the first kind (RF).
- elliprg: Symmetric elliptic integral of the second kind (RG).
- elliprj: Symmetric elliptic integral of the third kind (RJ).
- elliprc: Degenerate elliptic integral of RF (RC).
- elliprd: Degenerate elliptic integral of the third kind (RD).
Miscellaneous functions
- jacobi_zeta: Jacobi Zeta function (Z).
- heuman_lambda: Heuman Lambda function (Λ0).

Testing

In the unit tests, the functions are tested against the Boost Math and Wolfram test data. Since Ellip accepts the argument m (parameter) instead of k (modulus) to allow larger domain support, the full accuracy report uses exclusively the Wolfram data. The full accuracy report, test data, and test generation scripts can be found here. The performance benchmark is presented to provide comparison between functions in Ellip. Comparing performance with other libraries is non-trivial, since they accept different domains.

Benchmark on AMD Ryzen 5 4600H with Radeon Graphics running x86_64-unknown-linux-gnu rustc 1.89.0 using ellip v0.4.0 at f64 precision (ε≈2.22e-16).

Legendre's Elliptic Integrals

Function	Median Error (ε)	Max Error (ε)	Mean Performance
ellipk	0.00	108.14	14.6 ns
ellipe	0.00	3.00	13.2 ns
ellipf	0.66	7.47	103.3 ns
ellipeinc	0.70	24.66	164.6 ns
ellippi	0.53	36.35	167.0 ns
ellippiinc	0.78	1.04e3	277.6 ns
ellippiinc_bulirsch	0.83	1.04e3	223.6 ns
ellipd	0.60	2.64	29.9 ns
ellipdinc	1.00	8.38	103.1 ns

Bulirsch's Elliptic Integrals

Function	Median Error (ε)	Max Error (ε)	Mean Performance
cel	0.70	38.34	34.0 ns
cel1	0.00	8.68	11.4 ns
cel2	0.61	3.97	22.8 ns
el1	0.00	1.60	37.8 ns
el2	0.70	79.92	56.7 ns
el3	0.70	46.32	117.5 ns

Carlson's Symmetric Integrals

Function	Median Error (ε)	Max Error (ε)	Mean Performance
elliprf	0.00	1.75	45.6 ns
elliprg	0.00	2.45	98.8 ns
elliprj	0.67	5.42e7	165.3 ns
elliprc	0.00	2.82	22.8 ns
elliprd	0.62	6.49	74.8 ns

Miscellaneous Functions

Function	Median Error (ε)	Max Error (ε)	Mean Performance
jacobi_zeta	1.42	9.83	242.4 ns
heuman_lambda	0.62	8.89	372.7 ns

Learn more at docs.rs.