How to use pgf-math inside an argument?

Question

I response to xport's recent questions, I tried to do the following

\usepackage{pgf}
[...]
\rule{\pgfmathdivide{4}{3}\pgfmathresult mm}{5mm}

Unfortunately, this fails with the following error message:

! Missing number, treated as zero.
<to be read again> 
                   \begingroup 
l.11 ...pgfmathdivide{4}{3}\pgfmathresult mm}{5mm}

My first guess was that for some reason the output of \pgfmathresult cannot be used as the input for \rule. However, the slight rearrangement

\pgfmathdivide{4}{3}\rule{\pgfmathresult mm}{5mm}

works as expected. Why? What do I have to change to be able to put \pgfmathdivide inside the argument of \rule (or rather inside a macro that can be used in the argument of \rule and similar commands)?

Answer 1

\pgfmathdivide is simply not expandable. By that, I mean it cannot run entirely in TeX's expansion processor.

A simple example of this is trying to assign to a register inside the replacement text of an \edef.

\edef\foo{\dimen0=3pt}
\show\foo

gives

> \foo=macro:
->\dimen 0=3pt.

Note that nothing actually expanded. The same principle is at work here. \pgfmathdivide expands to at least one assignment (to \pgfmathresult) and since assignments are not expandable, \pgfmathdivide cannot be used in an expansion context (at least not the way you want).

It might be possible to reimplement division in a way that is expandable, for example using ε-TeX's expression primitives, but it would probably be a nontrivial amount of work.

Answer 2

As TH notes in his answer, the macros from pgfmath are not expandable. In fact, I rewrote the l3fp package from scratch a year ago (well, it took me many months) to provide an expandable version of what many other packages provide. Writing expandable code means in particular that it is not possible to store results in variables, and of course that makes it very tricky to parse expressions, or compute trigonometric functions. Long story short, we can now perform computations expandably, hence the following works.

\documentclass{article}
\usepackage{expl3}
\ExplSyntaxOn
\cs_set_eq:NN \eval \fp_eval:n
\ExplSyntaxOff
\begin{document}
  \rule{\eval{4/3}mm}{5mm}
\end{document}

First make the programming command \fp_eval:n available at document level, as \eval, then use it in the argument of \rule. The \fp_eval:n command takes one argument, evaluates it as a floating point expression, and produces a decimal number, so after expansion the argument of \rule is 1.333333333333333mm (16 significant figures). The l3fp package also provides other functions, for instance \fp_to_scientific:n, which would give things like 3.4e5, or \fp_to_tl:n, which produces a scientific representation for very large or very small numbers, but otherwise a decimal representation. Another goal of l3fp is to follow the (decimal) IEEE 854 (now absorbed into 754) standard. A major omission is the absence of support for subnormal numbers.