These functions are represented as LaTeX, because that's the gold standard for rendering math. But, they also want them available as XML so they're in a more sensible format for archiving.

So, funnily, as part of this, NIST maintain a LaTeX to XML converter.[0] It's a pretty complete LaTeX runtime. (Yes, runtime -- TeX is a programming language, not a markup language!)

This is the thing that powers arXiv Vanity[1] and other academic publishing projects.

I just find it quite amusing that the US Government maintains a LaTeX runtime written in Perl.

[0] https://dlmf.nist.gov/LaTeXML/

[1] https://www.arxiv-vanity.com/

> This is the thing that powers arXiv Vanity.

Never heard of this. Does it have access to the LaTeX source of the arXiv paper? I didn't know arXiv hosted sources.

Depends what the authors upload. You can just upload a PDF. But if arXiv detects that the PDF came out of pdflatex it will pretty instantly nag you to give it the TeX sources. It will then render the PDF itself and also offer the sources for download. That is also how astro-ph-leaks [1] works.

[1] https://twitter.com/LeaksPh

Oh, wow. Had never heard of astro-ph-leaks. That is the best.

Xml in what sense? Mathml?

It's own format, which is a rough semantic representation of a LaTeX document: https://dlmf.nist.gov/LaTeXML/manual/schema/

There are ways of exporting MathML within that schema, I think.

The site has MathML (pMML) representations that can be downloaded.

It reminds me of Rubi [0], a rule-based symbolic integration system. Unlike many CASs that use a variant of the Risch algorithm, Rubi simply solves the integrals by using a large number (6700+) of rules in its database. Benchmarks (72000+ test cases) show that it's able to get better and cleaner antiderivatives than Mathematica and Maple. All the rules [1] are both human and machine-readable.

The engine can be installed in Mathematica, SymPy and SymJa.

[0] https://rulebasedintegration.org/

[1] https://rulebasedintegration.org/integrationRules.html

See also FunGrim: https://fungrim.org/

> The Mathematical Functions Grimoire (Fungrim) is an open source library of formulas and data for special functions. Fungrim currently consists of 456 symbols (named mathematical objects), 3130 entries (definitions, formulas, tables, plots), and 82 topics (listings of entries).

"FunGrim: a symbolic library for special functions" https://arxiv.org/abs/2003.06181

This is great, but I wish there were a broader library that went beyond special functions.

For example, last year I was doing some statistical modelling and ended up with a complicated integral that yielded to analysis much more easily than I expected. I wanted to see if the resulting formula had a name or had been studied by anyone, but it's really hard to search for something like that. Anyone who did that analysis previously might have used different variable names or different conventions to represent the same underlying idea, factored it differently, etc.

Will the intersection of deep learning and differential geometry obviate the need for Euclidean representations altogether? It would be amazing if these functions became almost like fossils. Historical curiosities of a bygone civilization ;)

Differential geometry for deep learning

https://metacademy.org/roadmaps/rgrosse/dgml

What do you mean by Euclidean representation?

Tensors. Triangle meshes in rendering. All the comfortable human abstractions we come to rely on. But which can be inefficient in high dimensional real world data

https://dawn.cs.stanford.edu/2019/10/10/noneuclidean/

How is any of that relevant to libraries of mathematical functions and (more generally) of mathematical models and their properties?

This article is mostly about scalar functions.

Did you look somewhere like the CRC Standard Mathematical Tables and Formulas book?

Try posting it on Crossvalidated: https://stats.stackexchange.com/

Or on Math Stackexchange: https://math.stackexchange.com/