# formal syntax rules of \dimexpr \numexpr \glueexpr

After looking at various answers on the site and by looking at the documentation of `e-tex`, I am still wondering what is exactly allowed in the various types of expressions, i.e. `\numexpr`, `\dimexpr` and `\glueexpr`.

More precisely, I am interested in where one could place real number division and multiplication. For example, is `\dimexpr 12pt / 2.3pt\relax` or something similar allowed? It could be helpful if one could explain the formal syntax of such expressions.

• The way to write your example is `\numexpr \dimexpr12pt / \dimexpr2.3pt`. But you get 5, while you probably wanted 5.2, since the division rounds to the nearest integer. (Not to be confused with \divide, which truncates instead of rounding.) Did you check out section 3.5 of the manual? In any case, if you want real division, there exist packages that do what you want. – Mark May 19 '15 at 2:30
• @Mark - I thought \divide simply did not allow division by a dimen, or is that just for plain tex? – John Kormylo May 19 '15 at 4:35
• @JohnKormylo You can't use `\divide` with a dimension but in `\numexpr \dimexpr12pt / \dimexpr2.3pt` TeX coerces the dimensions into integers in scaled points then does the number expression division (which rounds as already noted). – Joseph Wright May 19 '15 at 7:27

The four expression primitives (`\numexpr`, `\dimexpr`, `\glueexpr`, `\muexpr`) can be used anywhere that a register of the same type can be. This means that `\numexpr` produces integers, `\dimexpr` produces dimensions, `\glueexpr` produces glue expressions and `\muexpr` produces muskip values. As such, something like the example in the question

``````\the\dimexpr 12pt / 2.3pt\relax
``````

cannot yield a real number as the result has to be a dimension. (As we'll see below, the result here is `6pt` followed by `.3pt\relax`.) (I've added `\the` here so you can typeset the result rather than having to use it in an assignment. I'll do that throughout.)

In all cases, the formal syntax is that the expressions absorb tokens and carry out appropriate mathematics up to either a `\relax` or the first non-valid token in the input. This is easiest to see with `\numexpr`, which must produce an integer. So for example

``````\the\numexpr 2 + 2pt
``````

will print `4pt`, which is made up of the result of the expression (`4`) followed by input which cannot be part of the expression and thus terminates it. This is useful in comparisons, where the comparator will necessarily terminate the expression

``````\ifnum\numexpr ... > \numexpr ... \relax
``````

In the case of `\dimexpr` and `\glueexpr`, the formal syntax requires that the input consists of a dimension (or glue value) followed by an optional multiplication factor, and not the other way around. The factor may itself be a numerical expression. Thus all of

``````\the\dimexpr 1pt + 1pt\relax
\the\dimexpr 1pt * 10\relax
\the\dimexpr (1pt + 1pt) / (1 + 1)\relax
``````

are valid and give the expected results, but

``````\the\dimexpr 10 * 1pt\relax
``````

is not valid and leads to an error.

Expressions may use the operators `+`, `-`, `*` and `/` along with parenthesis `(`/`)`. Division takes place with rounding, and this applies to any number expressions inside dimension/glue expressions. Thus for example we get

``````\the\dimexpr 5pt * (3 / 2) \relax % 10pt
\the\dimexpr 5pt * (4 / 3) \relax % 5pt
``````

In glue expressions the plus/minus parts do not need parenthesis to be affected by a factor as they are necessarily one entity. So for example

``````\the\glueexpr 5pt plus 1pt * 2 \relax
``````

yields `10pt plus 2pt`.

Within expressions, TeX will coerce other numerical types into the correct form in the same way as occurs when doing register assignment. So

``````\the\numexpr\dimexpr 1pt\relax\relax
``````

will result in `65536`, i.e. the value of `1pt` converted to scaled points and then coerced into an integer. With `\glueexpr` here the stretch/shrink would be dropped. Going the other way, any `\numexpr` inside a `\dimexpr` or `\glueexpr` will need appropriate units providing

``````\the\dimexpr\numexpr 1 + 2\relax pt\relax
``````
• `\muexpr` is feeling left out – David Carlisle May 19 '15 at 11:04