blog-static/content/blog/backend_math_rendering.md

---
title: Rendering Mathematics On The Back End
date: 2020-07-21T14:54:26-07:00
tags: ["Website", "Nix", "Ruby", "KaTeX"]
---

Due to something of a streak of bad luck when it came to computers, I spent a
significant amount of time using a Linux-based Chromebook, and then a
Pinebook Pro. It was, in some way, enlightening. The things that I used to take
for granted with a 'powerful' machine now became a rare luxury: StackOverflow,
and other relatively static websites, took upwards of ten seconds to finish
loading. On Slack, each of my keypresses could take longer than 500ms to
appear on the screen, and sometimes, it would take several seconds. Some
websites would present me with a white screen, and remain that way for much
longer than I had time to wait. It was awful.

At one point, I installed uMatrix, and made it the default policy to block
all JavaScript. For the most part, this worked well. Of course, I had to
enable JavaScript for applications that needed to be interactive, like
Slack, and Discord. But for the most part, I was able to browse the majority
of the websites I normally browse. This went on until I started working
on the [compiler series]({{< relref "00_compiler_intro.md" >}}) again,
and discovered that the LaTeX math on my page, which was required
for displaying things like inference rules, didn't work without
JavaScript. I was left with two options:

* Allow JavaScript, and continue using MathJax to render my math.
* Make it so that the mathematics are rendered on the back end.

I've [previously written about math rendering]({{< relref "math_rendering_is_wrong.md" >}}),
and made the observation that MathJax's output for LaTeX is __identical__
on every computer. From the MathJax 2.6 change log:

> _Improved CommonHTML output_. The CommonHTML output now provides the same layout quality and MathML support as the HTML-CSS and SVG output. It is on average 40% faster than the other outputs and the markup it produces are identical on all browsers and thus can also be pre-generated on the server via MathJax-node.

It seems absurd, then, to offload this kind of work into the users, to
be done over and over again. As should be clear from the title of
this post, this made me settle for the second option: it was
__obviously within reach__, especially for a statically-generated website
like mine, to render math on the backend.

I settled on the following architecture:

* As before, I would generate my pages using Hugo.
* I would use the KaTeX NPM package to render math.
* To build the website no matter what system I was on, I would use Nix.

It so happens that Nix isn't really required for using my approach in general.
I will give my setup here, but feel free to skip ahead.

### Setting Up A Nix Build
My `default.nix` file looks like this:

```Nix {linenos=table}
    { stdenv, hugo, fetchgit, pkgs, nodejs, ruby }:

    let
        url = "https://dev.danilafe.com/Web-Projects/blog-static.git";
        rev = "<commit>";
        sha256 = "<hash>";
        requiredPackages = import ./required-packages.nix {
            inherit pkgs nodejs;
        };
    in
        stdenv.mkDerivation {
            name = "blog-static";
            version = rev;
            src = fetchgit {
                inherit url rev sha256;
            };
            builder = ./builder.sh;
            converter = ./convert.rb;
            buildInputs = [
                hugo
                requiredPackages.katex
                (ruby.withPackages (ps: [ ps.nokogiri ]))
            ];
        }
```

I'm using `node2nix` to generate the `required-packages.nix` file, which allows me,
even from a sandboxed Nix build, to download and install `npm` packages. This is needed
so that I have access to the `katex` binary at build time. I fed the following JSON file
to `node2nix`:

```JSON {linenos=table}
[
    "katex"
]
```

The Ruby script I wrote for this (more on that soon) required the `nokogiri` gem, which
I used for traversing the HTML generated for my site. Hugo was obviously required to
generate the HTML.

### Converting LaTeX To HTML
After my first post complaining about the state of mathematics on the web, I received
the following email (which the author allowed me to share):

> Sorry for having a random stranger email you, but in your blog post
[(link)]({{< relref "math_rendering_is_wrong" >}}) you seem to focus on MathJax's
difficulty in rendering things server-side, while quietly ignoring that KaTeX's front
page advertises server-side rendering. Their documentation [(link)](https://katex.org/docs/options.html)
even shows (at least as of the time this email was sent) that it renders both HTML
(to be arranged nicely with their CSS) for visuals and MathML for accessibility.

The author of the email then kindly provided a link to a page they generated using KaTeX and
some Bash scripts. The math on this page was rendered at the time it was generated.

This is a great point, and KaTeX is indeed usable for server-side rendering. But I've
seen few people who do actually use it. Unfortunately, as I pointed out in my previous post on the subject,
few tools actually take your HTML page and replace LaTeX with rendered math.
Here's what I wrote about this last time:

> [In MathJax,] The bigger issue, though, was that the `page2html`
program, which rendered all the mathematics in a single HTML page,
was gone. I found `tex2html` and `text2htmlcss`, which could only
render equations without the surrounding HTML. I also found `mjpage`,
which replaced mathematical expressions in a page with their SVG forms.

This is still the case, in both MathJax and KaTeX. The ability
to render math in one step is the main selling point of front-end LaTeX renderers:
all you have to do is drop in a file from a CDN, and voila, you have your
math. There are no such easy answers for back-end rendering. In fact,
as we will soon see, it's not possible to just search-and-replace occurences
of mathematics on your page, either. To actually get KaTeX working
on the backend, you need access to tools that handle the potential variety
of edge cases associated with HTML. Such tools, to my knowledge, do not
currently exist.

I decided to write my own Ruby script to get the job done. From this script, I
would call the `katex` command-line program, which would perform
the heavy lifting of rendering the mathematics.

There are two types of math on my website: inline math and display math.
On the command line ([here are the docs](https://katex.org/docs/cli.html)),
the distinction is made using the `--display-mode` argument. So, the general algorithm
is to replace the code inside the `$$...$$` with their display-rendered version,
and the code inside the `\(...\)` with the inline-rendered version. I came up with
the following Ruby function:

```Ruby {linenos=table}
def render_cached(cache, command, string, render_comment = nil)
  cache.fetch(string) do |new|
    puts "  Rendering #{render_comment || new}"
    cache[string] = Open3.popen3(command) do |i, o, e, t|
      i.write new
      i.close
      o.read.force_encoding(Encoding::UTF_8).strip
    end
  end
end
```

Here, the `cache` argument is used to prevent re-running the `katex` command
on an equation that was already rendered before (the output is the same, after all).
The `command` is the specific shell command that we want to invoke; this would
be either `katex` or `katex -d`. The `string` is the math equation to render,
and the `render_comment` is the string to print to the console instead of the equation
(so that long, display math equations are not printed out to standard out).

Then, given a substring of the HTML file, we use regular expressions
to find the `\(...\)` and `$$...$$`s, and use the `render_cached` method
on the LaTeX code inside.

```Ruby {linenos=table}
def perform_katex_sub(inline_cache, display_cache, content)
  rendered = content.gsub /\\\(((?:[^\\]|\\[^\)])*)\\\)/ do |match|
    render_cached(inline_cache, "katex", $~[1])
  end
  rendered = rendered.gsub /\$\$((?:[^\$]|$[^\$])*)\$\$/ do |match|
    render_cached(display_cache, "katex -d", $~[1], "display")
  end
  return rendered
end
```

There's a bit of a trick to the final layer of this script. We want to be
really careful about where we replace LaTeX, and where we don't. In
particular, we _don't_ want to go into the `code` tags. Otherwise,
it wouldn't be possible to talk about LaTeX code! I also suspect that
some captions, alt texts, and similar elements should also be left alone.
However, I don't have those on my website (yet), and I won't worry about
them now. Either way, because of the code tags,
we can't just search-and-replace over the entire page; we need to be context
aware. This is where `nokogiri` comes in. We parse the HTML, and iterate
over all of the 'text' nodes, calling `perform_katex_sub` on all
of those that _aren't_ inside code tags.

Fortunately, this kind of iteration is pretty easy to specify thanks to something called XPath.
This was my first time encountering it, but it seems extremely useful: it's
a sort of language for selecting XML nodes. First, you provide an 'axis',
which is used to specify the positions of the nodes you want to look at
relative to the root node. The axis `/` looks at the immediate children
(this would be the `html` tag in a properly formatted document, I would imagine).
The axis `//` looks at all the transitive children. That is, it will look at the
children of the root, then its children, and so on. There's also the `self` axis,
which looks at the node itself.

After you provide an axis, you need to specify the type of node that you want to
select. We can write `code`, for instance, to pick only the `<code>....</code>` tags
from the axis we've chosen. We can also use `*` to select any node, and we can
use `text()` to select text nodes, such as the `Hello` inside of `<b>Hello</b>`.

We can also apply some more conditions to the nodes we pick using `[]`.
For us, the relevant feature here is `not(...)`, which allows us to
select nodes that do __not__ match a particular condition. This is all
we need to know.

We write:

* `//`, starting to search for nodes everywhere, not just the root of the document.
* `*`, to match _any_ node. We want to replace math inside of `div`s, `span`s, `nav`s,
all of the `h`s, and so on.
* `[not(self::code)]`, cutting out all the `code` tags.
* `/`, now selecting the nodes that are immediate descendants of the nodes we've selected.
* `text()`, giving us the text contents of all the nodes we've selected.

All in all:

```
//*[not(self::code)]/text()
```

Finally, we use this XPath from `nokogiri`:

```Ruby {linenos=table}
files = ARGV[0..-1]
inline_cache, display_cache = {}, {}

files.each do |file|
  puts "Rendering file: #{file}"
  document = Nokogiri::HTML.parse(File.open(file))
  document.search('//*[not(self::code)]/text()').each do |t|
    t.replace(perform_katex_sub(inline_cache, display_cache, t.content))
  end
  File.write(file, document.to_html)
end
```

I named this script `convert.rb`; it's used from inside of the Nix expression
and its builder, which we will cover below.

### Tying it All Together
Finally, I wanted an end-to-end script to generate HTML pages and render the LaTeX in them.
I used Nix for this, but the below script will largely be compatible with a non-Nix system.
I came up with the following, commenting on Nix-specific commands:

```Bash {linenos=table}
# Nix-specific; set up paths.
source $stdenv/setup

# Build site with Hugo
# The cp is Nix-specific; it copies the blog source into the current directory.
cp -r $src/* .
hugo --baseUrl="https://danilafe.com"

# Render math in HTML and XML files.
# $converter is Nix-specific; you can just use convert.rb.
find public/ -regex "public/.*\.html" | xargs ruby $converter

# Output result
# $out is Nix-specific; you can replace it with your destination folder.
mkdir $out
cp -r public/* $out/
```

This is it! Using the two scripts, `convert.rb` and `builder.sh`, I
was able to generate my blog with the math rendered on the back-end.
Please note, though, that I had to add the KaTeX CSS to my website's
`<head>`.

### Caveats
The main caveat of my approach is performance. For every piece of
mathematics that I render, I invoke the `katex` command. This incurs
the penalty of Node's startup time, every time, and makes my approach
take a few dozen seconds to run on my relatively small site. The
better approach would be to use a NodeJS script, rather than a Ruby one,
to perform the conversion. KaTeX also provides an API, so such a NodeJS
script can find the files, parse the HTML, and perform the substitutions.
I did quite like using `nokogiri` here, though, and I hope that an equivalently
pleasant solution exists in JavaScript.

Re-rendering the whole website is also pretty wasteful. I rarely change the
mathematics on more than one page at a time, but every time I do so, I have
to re-run the script, and therefore re-render every page. This makes sense
for me, since I use Nix, and my builds are pretty much always performed
from scratch. On the other hand, for others, this may not be the best solution.

### Alternatives
The same person who sent me the original email above also pointed out
[this `pandoc` filter for KaTeX](https://github.com/Zaharid/pandoc_static_katex).
I do not use Pandoc, but from what I can see, this fitler relies on
Pandoc's `Math` AST nodes, and applies KaTeX to each of those. This
should work, but wasn't applicable in my case, since Hugo's shrotcodes
don't mix well with Pandoc. However, it certainly seems like a workable
solution.

### Conclusion
With the removal of MathJax from my site, it is now completely JavaScript free,
and contains virtually the same HTML that it did beforehand. This, I hope,
makes it work better on devices where computational power is more limited.
I also hope that it illustrates a general principle - it's very possible,
and plausible, to render LaTeX on the back-end for a static site.