Using operator spacing to indicate precedence

Question

I regularly have to typeset equations that are, because of their inherent complexity, difficult to read. To address this I have gotten into the habit of emphasizing operator precedence by horizontal spacing.

To give a simple example, because in mathematics \times binds stronger than +, and + binds stronger than =, I would do things like this:

\[  a \times b \; + \; c \times d \;\; = \;\; (e + f) \times g  \]

I've found that this immediately makes it easier to understand an equation at a glance.

Well, since then I have devised a way that is a little nicer than that, defining a family of custom 'math ligatures' using the semantic package:

\[  a `*` b ``+`` c `*` d ```=``` (e + f) `*` g  \]

Increasing the number of backticks exponentially increases the extra space, and I've experimentally come up with lengths that appear most pleasing to the eye. The implementation comes down to the following (though I have left out a number of corner cases and features, such as encoding \times as `*`):

\RequirePackage [ligature] {semantic}

\mathlig{`````}{\mspace{38mu}}
\mathlig{````}{\mspace{23mu}}
\mathlig{```}{\mspace{12mu}}
\mathlig{``}{\mspace{5mu}}

Have other people already attempted something similar? Is there a way to do this better / easier / automatically?

---

On a more philosophical note (and I hope this doesn't violate the spirit of stackexchange; mods, please let me know if I should remove this), I somehow expect this to be a controversial approach. After all, many die-hard TeX users go through great lengths to preserve the beautiful spacing that TeX generates. I would actually like to hear from you. How do you solve the problem of formula readability?

Answer 1

If your scope is limited to the main arithmetical ones, that is, plus, minus, times and equals, you can do the same without any markup.

Just define the characters +, -, * and = to be math active.

Here's a Plain TeX document, the same will do with LaTeX (if you load amsmath the declarations \mathcode`+="8000 and similar ones should be inside the argument to \AtBeginDocument).

\mathchardef\plus=\mathcode`+
\mathchardef\minus=\mathcode`-
\mathchardef\equals=\mathcode`=

\begingroup\lccode`~=`+
\lowercase{\endgroup\def~}{\mkern12mu\plus\mkern12mu}
\begingroup\lccode`~=`-
\lowercase{\endgroup\def~}{\mkern12mu\minus\mkern12mu}
\begingroup\lccode`~=`*
\lowercase{\endgroup\def~}{\mkern5mu\times\mkern5mu}
\begingroup\lccode`~=`=
\lowercase{\endgroup\def~}{\mkern23mu\equals\mkern23mu}

\mathcode`+="8000
\mathcode`-="8000
\mathcode`*="8000
\mathcode`=="8000

$$
a * b + c * d = (e * f) - g
$$
\bye

Of course in LaTeX one should use \[...\] instead of $$...$$.

You can also activate this "on demand":

\mathchardef\plus=\mathcode`+
\mathchardef\minus=\mathcode`-
\mathchardef\equals=\mathcode`=

\def\spacedmath{%
  \begingroup\lccode`~=`+
  \lowercase{\endgroup\def~}{\mkern12mu\plus\mkern12mu}
  \begingroup\lccode`~=`-
  \lowercase{\endgroup\def~}{\mkern12mu\minus\mkern12mu}
  \begingroup\lccode`~=`*
  \lowercase{\endgroup\def~}{\mkern5mu\times\mkern5mu}
  \begingroup\lccode`~=`=
  \lowercase{\endgroup\def~}{\mkern23mu\equals\mkern23mu}
  \mathcode`+="8000
  \mathcode`-="8000
  \mathcode`*="8000
  \mathcode`=="8000
}

$$\spacedmath
a * b + c * d = (e * f) - g
$$
$$
a * b + c * d = (e * f) - g
$$
\bye

The usefulness of this from a pedagogical point of view is dubious.