How to do arithmetic without assignment?

Question

I have known that I can use "advance...by..." to do arithmetic on number in TeX, but is there any way to do arithmetic without changing the value of number? For example, if I want to determine whether a number called X whose value is x satisfies x^2+x+1<0, and keep the former value of X for future use, what can I do?

Answer 1

You can e.g. use xfp:

\documentclass{article}
\usepackage{xfp}
\newcommand\myxvalue{15}

\begin{document}

\ifnum \fpeval{\myxvalue^2 + \myxvalue + 1} < 0 \relax 
        yes  
\else no: it is \fpeval{\myxvalue^2 + \myxvalue + 1} 
\fi

\end{document}

Answer 2

The xfp package is wrapper to access at user level the "Floating Point Unit" of LaTeX3. All its operations are expandable.

At the core level they use crucially the e-TeX addition to Knuth TeX of \numexpr, \dimexpr... which allow expandable computations with infix notations (and some sometimes surprising restrictions).

using directly \numexpr/\dimexpr might be all you need.

Maybe you need computations with integers bigger than 2^31 and you want ^operator? and factorial with !? then you can use bnumexpr.

For even more complicated things like add(x(x+1), x=1..100) you can use xintexpr.

An earlier package allowing expandable computations inclusive of powers was intcalc but without infix notations, only macros. Also bigintcalc does it with integers exceeding the 2^31 bound (intcalc achieves expandable computations even if the TeX-engine is lacking e-TeX \numexpr; this is interesting theoretically but quite slower in practice. It does use \numexpr if the latter is available).

As mentioned above xintexpr has extended infix syntax (and is faster when computing with "big" numbers). bnumexpr is a more light-weight package, which hooks also into the core arithmetics provided by xintcore, but does not develop as extensive a syntax as xintexpr.

Naturally at some point you will want to actually store the computations somewhere, via an \edef for example; so the computations are done expandably, without assignments, which is nice for \ifnum, for \write etc... (for embedding in some TikZ things, too), but at end of day you almost always have to do something unexpandable...

Then, things like fp, pgfmath, pgfplots, apnum, are math engine doing things non-expandably to start with (maybe I forget some, please complete). The apnum is arbitrary precision.

As per your original question, perhaps something in this style:

\xintdefvar X := whatever with big integers and fractions;
% 
\xintifboolexpr{X^2 + X + 1 < 0}{Yes branch}{No branch}

The first steps does the computations expandably and defines variable X. Then \xintifboolexpr is purely expandable and evaluates X^2 + X + 1 with no assignments.

Of course as was observed by @Skillmon always the No branch is taken here... (there is no interface to complex numbers in xintexpr so far).