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Recent work by Nelson H. F. Beebe
Nelson H. F. Beebe
University of Utah
Department of Mathematics, 110 LCB
155 S 1400 E RM 233
Salt Lake City, UT 84112-0090
USA
Email: beebe@math.utah.edu, beebe@acm.org, beebe@computer.org
ORCID: 0000-0001-7281-4263
WWW URL: https://www.math.utah.edu/~beebe
Telephone: +1 801 581 5254

Original version: Wed Oct 28 16:22:09 2026

Update(s): Wed Oct 28 16:22:09 2026     …     Tue Nov 3 13:31:02 2026

This Web site collects short summaries of work that I have done in the last few years, with links to more details, including ones that represent years of software development, and have detailed supporting technical books. My interest has always been in doing a thorough job of documenting my work, and making it freely accessible to the world (apart from book publication, which is often a commercial and copyrighted business).

• Algol 68 at Utah: I am targeting this new compiler for an old language with a proposed interface from that language to a large body of existing libraries for scientific computation, most notably, for my MathCW library treated later on this page.
• CORDIC algorithms: These are interesting computational routes to several different mathematical functions that, at the lowest level, need only integer shift and add operations, making them suitable for fast hardware implementation. My project investigates their applicability for implementation in high-level programming languages, and the accuracy attainable. Even if they prove to be inferior to other methods for reasons of accuracy or execution time, they are still worthy of investigation, because they provide radically different routes to, and completely independent computation of, function values, making them invaluable for software testing.
• Decimal-arithmetic support in gcc compilers: This describes extensions to the GNU gcc compiler family to allow C code to use decimal floating-point arithmetic. My MathCW library discussed later provides the entire software layer above the compiler-supported basic decimal operations (add, subtract, multiply, and divide) that is needed to raise decimal arithmetic to the level of existing support for binary arithmetic. With care, code in C and C++ can be written to be largely independent of precision and base.
• Fixed-point binary arithmetic library: fx. This project has not yet reached public release, but has a lot of work in it, including a library of 652 functions in 120_000 lines of highly portable C code, and a book of viii + 343 pages, including a detailed index of 43 pages, a 126-entry bibliography, and 4-page author/editor index linking every author or editor to the locations in the book where their work is cited. The library builds on the lessons of the IEEE 754 floating-point arithmetic system, but extends that to high-precision fixed-point arithmetic where addition and subtraction are always exact, and where several additional rounding modes are available. The library supplies all of the numerical functions specified by the ISO C, C++, and Fortran Standards, and includes several more special functions, including cumulative distribution functions and their inverses, incomplete gamma functions, exponential integral functions, logarithmic integrals, regular and hyperbolic cosine and sine exponential integrals, chi-square probability functions and their inverses, dilogarithm, polylogarithms, and a collection of utility functions for bit operations, high-resolution timers, hash functions, and hardware and operating-system random number generators. One chapter shows how fixed-point arithmetic can be as easy to use in C++ as normal arithmetic is, through type abstractions and operator overloading. The book also addresses the needs of high-precision arithmetic in cryptography, and its defense against timing attacks.
• hoc: The greatly extended High-Order Calculator. This is a long running project that now consists of 130_000 lines of portable C code that runs, and has been tested on, all of the major CPU types used in common computers since about 1990, and supports four binary floating-point formats and three decimal formats. It also has library interfaces to allow simulation of three variants of 16-bit floating-point arithmetic that are commonly used in AI and LLM modeling. Hoc allows run-time loading of interfaces to dynamic load libraries, allowing access to math libraries beyond the traditional one for C, C++, and Fortran. All of its functions have online documentation from inside a hoc session, as well as support for searches in that documentation, and for pattern matching of its symbol table of about 900 functions, procedures, and named constants. It has recently gained low-level and high-level interfaces to the MPFR library for correctly rounded multiple-precision arithmetic. Hoc is used extensively in the MathCW book discussed later on this Web page to illustrate small computations without the clutter required in C and C++. An earlier version of Hoc is documented in a 2008 edition of a free book in color (xxiv + 363 pages), but there is much more beyond that now, and time has yet to be found for an extended edition. Hoc has proved to be a valuable tool for experimenting with, learning about, and teaching, computer arithmetic, and also for prototyping algorithms that might later be reimplemented in other programmming languages
• Kefir C compiler for x86_64. Through my urging, this recently developed compiler from an expert programmer in Europe, has recently been extended with support for decimal floating-point arithmetic, and I am routinely using it as an independent alternative to dgcc for decimal code development.
• MathCW Project: This multi-year work produced a large published book, The Mathematical Function Computation Handbook (Springer, 2017, ISBN 3-319-64109-3, DOI: 10.1007/978-3-319-64110-2), and a huge library of mathematical software that includes, and greatly extends, the mathematical and I/O libraries required by ISO Standards for C, C++, and Fortran. It covers the 32-, 64-, 80-, and 128-bit IEEE 754 binary floating-point formats, the 32-, 64-, and 128-bit IEEE 754 decimal floating-point formats, and supports important historical floating-point formats (IBM System/360 hexadecimal, DEC PDP-10 binary, and DEC PDP-11 and VAX binary). This is all done from a single parametrized code base where most computational algorithms are carefully designed to be independent of both base and precision, and portable across numerous operating systems and diverse CPU types used in desktop computers since about 1990. In addition, the MathCW library is designed to support a future IEEE 754-like 256-bit format offering about 70 decimal digits in both binary and decimal formats, and should be usable as soon as at least one C compiler implements support for those two new formats. It is the only major handbook treatment of decimal floating-point arithmetic that deals with real implementation issues, and places programming in base 10 arithmetic on an equal footing with base 2 arithmetic. The MathCW book includes appendices on interfaces to the MathCW library from Ada, C++, C#, Fortran, Java, and Pascal. Since the book was published, several more interfaces have been developed, and are freely available via links in the next several items. An interface from Algol 68 is planned, once some interlanguage calling issues have been resolved in a new compiler for that language.
• MathCW book Virtual Appendix A68: Algol 68 interface: outlined and planned, but not yet written.
• MathCW book Virtual Appendix K: COBOL interface
• MathCW book Virtual Appendix G: Go interface
• MathCW book Virtual Appendix M: Modula-2 interface
• MathCW book Virtual Appendix O: Oberon interface
• MathCW book Virtual Appendix R: Rust interface
• Modula-2 code examples
• Profiling and extending awk: This late-2026 work hugely extends the Awk language to include access to the entire C89 and C99 math libraries, hexadecimal integer and floating-point constants, and supplies versions for four IEEE 754 binary formats, and three IEEE 754 decimal formats. They can be easily built on numerous Unix-family operating systems, with all of the CPU types in common use since about 1990. They make the wonderful Awk programming language applicable to analysis of numeric data from any of the IEEE 754 formats supported in C and other programming languages. Awk's abstraction of integer and floating-point types into a single type, number, that can be trivially converted to and from string representations makes some numeric programming much easier in pawk than in C, C++, Go, Rust, and many other languages.
• Pochhammer symbol computation: This small project investigates accurate computation of the Pochhammer symbol, also called the rising factorial. It is a critical component in the computation of the gamma function, factorials, and binomial coefficients. Its computation may appear trivial, with just repeated multiplications, but removing the growing error from those operations is hard. The project reports on several alternatives.
• Random-number generation and testing: This active site collects information about random number algorithms and their testing, and supplies portable implementations of several different generators, including two little-known ones that are of high quality, and deserve wider recognition.
• TeX Live 2025: this is part of an annual report series that started in 2016 to test the buildability of the TeX Live code bases on numerous operating systems, and CPU platforms. In some years, more than 100 systems have been used in that testing. I maintain the master North American mirrors for both TeX Live, and the Comprehensive TeX Archive Network (CTAN) collections. The first of those has about 188_000 files and 13_550 directories. The second has about 360_000 files and 19_000 directories. Both collections are freely accessible to the world, actively maintained, and mirrored at scores of sites around the world.
• TeX Live 2026
• TeX Live 2027
• Underflows in computer arithmetic: This is a collection of test software, and a PDF of a section from an in-progress book, about some little understood features of small numbers.
• William Kahan's minimal function kernels: This project implements and investigates a proposal by the chief architect of the IEEE 754 floating-point design for a minimal set of functions from which several others in the standard C, C++, and Fortran libraries can easily derived. In particular, it examines the accuracy attainable by that approach.
• University of Utah graduate document support:

  For more than three decades, a now-retired colleague and I tried to maintain an early-1990s vintage set of style files for typesetting University of Utah graduate theses and dissertations. That collection is available here, and has been in use since the 1990s. However, the approach taken by its original student developer cut off many avenues of improvement, and it badly needs to be retired.

  In 2018, I therefore wrote a completely new, and clean, interface that builds on top of standard LaTeX, and provides access to its thousands of style files, extended fonts, improved typography, multilingual typesetting, color, and hypertext links for Web access to PDF file navigation. That package is available for Department of Mathematics students here. It adds support for undergraduate honors theses, and provides document samples of increasing complexity, up to including published papers in chapters, supplying multiple levels of indexing, incorporating texts in other scripts in both left-to-right and right-to-left writing directions, and producing editions in larger font sizes to assist readers with vision difficulties.

  In the last two years, small extensions have been made to address demands for document accessibility, which means producing PDFs that are easier to navigate by humans with visual and/or physical disabilities. A package version for Fall 2026 is being tested by a few students, and will soon replace the previous version at the linked-to site.

• Bibliography archives:

  The long running BibNet Project and TeX User Group bibliography archives are continually expanded.

  The first has coverage of the works of important physical scientists, computer scientists, mathematicians, and statisticians, with a few subject-specific bibliographies as well. For most of them, these are the most comprehensive collections available anywhere, and are high-quality, and reliable, citation material.

  The second includes several subject-specific bibliographies, plus almost always complete coverage of hundreds of journals in computer science, computer engineering, fonts, numerical mathematics, probability, statistics, typography, and the history of mathematics and science. In collaboration with a research scientist at NOAA, it also includes coverage of about two dozen journals in fisheries research. While there is little overlap of those with the other bibliographies, our joint work exposed the need for additional markup practices in biological and geographic literature citations.

  Collectively, these bibliography archives hold more than 2.2 million entries, each of which has been subjected to numerous checks, and successfully typeset. They are searchable in individual author, journal, or subject files, but more powerfully are accessible through any of several SQL databases that allow precise subfield searching and ordering of results. The SQL interface is described in here, and some of the bibliographic software tools are documented here.

  These bibliography archives are mirrored at numerous places around the globe, including at Oak Ridge National Laboratory in Tennessee, and Sandia National Laboratory in Albuquerque, New Mexico. Their data are freely reusable by others, without restriction.