---
title: "Arb: Efficient Arbitrary-Precision Midpoint-Radius Interval Arithmetic"
description: "Johansson presents Arb, a portable, thread-safe C library built on midpoint-radius (\"ball\") arithmetic. 32 references, cited by 8 papers here."
authors: ["Fredrik Johansson"]
year: 2016
venue: "arXiv:1611.02831"
source: "https://arxiv.org/abs/1611.02831"
doi: "10.48550/arxiv.1611.02831"
arxiv: "1611.02831"
licence: "arXiv — non-exclusive licence to distribute"
licence_url: "https://arxiv.org/licenses/nonexclusive-distrib/1.0/license.html"
summary_licence: "Elamur Cosmic License"
canonical: "https://research.elamur.ai/free/en/papers/johansson-2016-arb-efficient-arbitrary-precision-midpoint/"
---
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# Arb: Efficient Arbitrary-Precision Midpoint-Radius Interval Arithmetic

**Authors:** Fredrik Johansson · **Year:** 2016 · **Venue:** arXiv:1611.02831 · **Licence:** [arXiv — non-exclusive licence to distribute](https://arxiv.org/licenses/nonexclusive-distrib/1.0/license.html)

**Links:** [Publisher's page](https://arxiv.org/abs/1611.02831) · [PDF](https://arxiv.org/pdf/1611.02831v1) · [DOI 10.48550/arxiv.1611.02831](https://doi.org/10.48550/arxiv.1611.02831) · [arXiv 1611.02831](https://arxiv.org/abs/1611.02831) · [OpenAlex](https://openalex.org/W2950825087)

**Citations:** 0 citations (OpenAlex, 2026-10-14) · [8](https://md.research.elamur.ai/free/en/papers/johansson-2016-arb-efficient-arbitrary-precision-midpoint/cited-by/) in our library, of the 149,439 papers whose bibliographies we read

## Summary
Rigorous numerical computation with interval arithmetic tracks error bounds automatically. Existing arbitrary-precision interval tools such as MPFI store both endpoints at full precision and offer only a narrow set of functions. Johansson presents Arb, a portable, thread-safe C library built on midpoint-radius ("ball") arithmetic. A real number is stored as [m ± r], where only the midpoint m needs full precision and the radius r needs just a few digits. At high precision this makes interval computation nearly as cheap as plain floating-point, and about twice as cheap as endpoint intervals. Arb builds real and complex balls, polynomials, power series and matrices from this core. It also provides rigorous evaluation of many special functions, including gamma, zeta, Bessel, Airy, hypergeometric and elliptic functions, over complex arguments. The paper covers the low-level representation, how precision and error bounds are managed, and fast polynomial arithmetic with interval coefficients. Performance is competitive with non-interval libraries such as MPFR and MPC. This makes certified results practical in computer algebra and scientific computing.

## Key points
- Arb's core types are arf_t (floating-point), mag_t (unsigned bounds), arb_t (real balls), acb_t (complex numbers), arb_poly_t/acb_poly_t (polynomials) and arb_mat_t/acb_mat_t (matrices). It depends on GMP/MPIR, MPFR and FLINT.
- As of November 2016 the library had about 1850 documented methods and roughly 650 test programs. Each test checks results for consistency on 10³–10⁶ random inputs, and the tests are also run under Valgrind.
- Arb is LGPL-licensed and can be used from SageMath, the python-flint module and Julia's Nemo package.
- A typical usage pattern is to repeat the computation with doubled precision until the target accuracy is reached. The paper demonstrates this by computing sin(π + e⁻¹⁰⁰⁰⁰) to 53 bits.
- The paper describes two applications:
  - About 500 lines of wrapper code give correctly rounded versions of all transcendental functions in MPFR 3.1.3.
  - A hydrogeophysics aquifer simulation uses Arb to obtain Bessel functions at 113-bit accuracy.

*Our summary of the paper, in our own words — not its text. The work belongs to its authors; read it at the links above.*

## Institutions

- [Université de Bordeaux](https://md.research.elamur.ai/free/en/universities/universite-de-bordeaux/) — University, Bordeaux, France · 2 works · 38,342 citations

Also on the byline: [Centre Inria de l'université de Bordeaux](https://ror.org/03tjcj052).

📚 Shelves (4): [🖥️ Computer Science](https://md.research.elamur.ai/free/en/shelves/computer-science/) · 19,503 works · [➗ Mathematics](https://md.research.elamur.ai/free/en/shelves/mathematics/) · 233 works · [🧮 Computational Theory and Mathematics](https://md.research.elamur.ai/free/en/shelves/computer-science/computational-theory-and-mathematics/) · 496 works · [➗ Modeling and Simulation](https://md.research.elamur.ai/free/en/shelves/mathematics/modeling-and-simulation/) · 12 works

## Cite

Cite as: Fredrik Johansson. 2016. Arb: Efficient Arbitrary-Precision Midpoint-Radius Interval Arithmetic. arXiv:1611.02831.

The paper itself, not this page. BibTeX from the library's record of the work.

```bibtex
@misc{johansson2016arb,
  title = {{Arb: Efficient Arbitrary-Precision Midpoint-Radius Interval Arithmetic}},
  author = {Fredrik Johansson},
  year = {2016},
  doi = {10.48550/arxiv.1611.02831},
  eprint = {1611.02831},
  archivePrefix = {arXiv},
  url = {https://doi.org/10.48550/arxiv.1611.02831}
}
```

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## References

32 references, 1 of them with a card of ours. ↗ leads to the work, «find» searches for one we have no id for yet.

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*Updated 3 November 2026*