Elliptic integrals for Rust

Ellip is a pure-Rust implementation of elliptic integrals. Ellip also provides less common functions like Bulirsch's cel and el. Some applications of the elliptic integrals include computing the lengths of plane curves, magnetism, astrophysics, and string theory.

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

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 can be found here, along with the test data and test generation scripts. 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 of input.

Benchmark on AMD Ryzen 5 4600H with Radeon Graphics @3.0 GHz running x86_64-unknown-linux-gnu rustc 1.90.0 using ellip v1.0.0 at f64 precision (ε=2.2204460492503131e-16).

Legendre's Elliptic Integrals

Function	Median Error (ε)	Max Error (ε)	Mean Performance
ellipk	0.00	108.14	14.8 ns
ellipe	0.00	3.00	13.1 ns
ellipf	0.00	7.47	98.5 ns
ellipeinc	0.00	24.66	157.6 ns
ellippi	0.00	36.35	166.1 ns
ellippiinc	0.00	395.31	232.9 ns
ellippiinc_bulirsch	0.00	395.31	189.2 ns
ellipd	0.00	2.64	30.0 ns
ellipdinc	0.00	8.38	98.9 ns

Bulirsch's Elliptic Integrals

Function	Median Error (ε)	Max Error (ε)	Mean Performance
cel	0.62	36.94	32.8 ns
cel1	0.00	8.68	11.2 ns
cel2	0.00	3.47	21.6 ns
el1	0.00	1.70	36.7 ns
el2	0.00	74.60	52.0 ns
el3	0.00	53.21	103.8 ns

Carlson's Symmetric Integrals

Function	Median Error (ε)	Max Error (ε)	Mean Performance
elliprf	0.00	1.57	46.1 ns
elliprg	0.00	5.25	99.1 ns
elliprj	0.56	136.97	165.5 ns
elliprc	0.00	2.82	22.5 ns
elliprd	0.00	6.25	75.9 ns

Miscellaneous Functions

Function	Median Error (ε)	Max Error (ε)	Mean Performance
jacobi_zeta	0.00	8.66	207.7 ns
heuman_lambda	0.00	2.86	333.8 ns

Reproducibility

This section describes how to reproduce the accuracy reports, test datasets, benchmarks, figures, and tables.

Setup the project

First, clone the repository:

git clone https://github.com/p-sira/ellip.git
cd ellip

Then, build the project:

cargo build --workspace

Run tests

For detailed information on tests, see tests/README.md.

Benchmark

Ellip's benchmark collects the test files associated with each function and reports the total execution time:

cargo bench

This produces raw benchmark output under target/criterion/. Note that the results shown in the README are normalized to per-function call averages. See the Generate tables section for details on generating the summary tables.

Generate tables

To generate the accuracy table in the test report:

cargo run --example generate_error_report

This compares Ellip's results against Wolfram test data and generates the accuracy report.

To generate the test and benchmark summary table as shown in the README, first run cargo bench to collect benchmark data. Then run:

cargo run --example generate_test_summary

This script compares results against Wolfram data, extracts benchmark results from target/criterion/, normalizes them to average time per function call, and summarizes everything in a single table.

Generate figures

To generate function plots:

cargo run -p ellip-plot-graph --bin [function-name]

See available plots in ellip-plot-graph/src/bin

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Learn more at docs.rs.