Simplify arithmetics +-, -+, --, 1x, -1x

Question

I made some command \pcoef and \mcoef that simplifies +- and -+ to -, and -- to + in arithmetic expressions, and a \coef that simplifies 1x and -1x to x and -x respectively. They are not really satisfying, since their use is a non natural arithmetic transcription.

I would like to produce +, - and * commands that could be use in a 'natural' way.

The following code contains the commands i made, and what i would like:

\documentclass[12pt,a4paper,notitlepage]{extarticle}
\usepackage{xfp} %for \fpeval,randint
\pagenumbering{gobble}
\setlength\parindent{0pt}
% the commands i made: not very natural use
\newcommand{\coef}[1]{\ifnum\numexpr#1=1\else\ifnum\numexpr#1=-1 -\else #1\fi\fi}
\newcommand{\pcoef}[1]{\ifnum\numexpr#1=1 + \else\ifnum\numexpr#1=-1 - \else\ifnum\numexpr#1>0 + #1 \else\ifnum\numexpr#1=0 + #1 \else #1\fi\fi\fi\fi}
\newcommand{\mcoef}[1]{\ifnum\numexpr#1=1 - \else\ifnum\numexpr#1=-1 + \else\ifnum\numexpr#1>0 - #1\else + \fpeval{-#1}\fi\fi\fi}
% the commands i would like:
\newcommand{+}{} % i don't need to move margin
\renewcommand{-}{}
\renewcommand{*}{} % i don't need line break in maths
\begin{document}
% data
% later \a, \b, \c, \d will take random values, positive or negative
\edef\a{ 15 }
\edef\b{ -17 }
\def\c{ 1 }
\def\d{ -1 }
{\bf what i can do, but with non natural arithmetic transcription } \[2mm]
$\a \pcoef{\b}$ \ % a+b : 15+-17 is changed to 15-17
$\a \mcoef{\b}$ \ % a-b : 15--17 is changed to 15+17
$\coef{\a} x \pcoef{\b}$ \ % ax+b : 15x+-17  is changed to 15x-17
$\coef{\b} x \pcoef{\a}$ \ % bx+a : -17x+15  is not changed
{\bf how to correct this, for a more natural arithmetics, like $\backslash$a$\backslash$+$\backslash$b, $\backslash$a$\backslash$-$\backslash$b, $\backslash$a$\backslash$*x$\backslash$+$\backslash$b,etc } \[2mm]
$\backslash$+:\ %+ :
$\a + \b =\ $ \leftarrow the $\backslash$+ instead of + should replace +- by -. \
$\backslash$-:\ %- :
$\a - \b =\ $ \leftarrow the $\backslash$- instead of - should replace -+ by -, and -- by +. \
$\backslash$*:\ %*
%* not sure if possible, since it might regcognize an arbitraby long integer before
% an alternative would therefore be *{c}x 
$\c * x =\ $\leftarrow the $\backslash$* instead of * should replace 1x by x. \
$\d  x =\ $\leftarrow the $\backslash$* instead of * should replace -1x by -x. \
$\a  x =\ $\leftarrow the $\backslash$* instead of * should replace a*x by ax if a is neither 1 or -1. \
\end{document}

Any help to achieve it would be appreciated.

Answer 1

With \regex_replace_all the documentation interface3.pdf

EDIT2:

• \newcommand{\a} causes an error, the command \a must already be defined elsewhere, I replace by \mya
• I created a 2nd parameter to hold the result of the simplication
• with the tutorial https://www.alanshawn.com/latex3-tutorial/#macro-expansion-control-v, replace \tl_set:Nn #1 { #2 } by \tl_set:Nx #1 { #2 }

EDIT3: with the comment of mathteacher, correction de \regex_replace_all:nnN { 1x } { x } #1 en \regex_replace_all:nnN { (^|[^\d])1([a-zA-Z\c{[A-Za-z]*}]+) } { \1\2 } #1

THE CODE

            \documentclass{article}
    %https://tex.stackexchange.com/questions/668419/simplify-arithmetics-1x-1x
    \ExplSyntaxOn
\NewDocumentCommand { \simp } { m m }
{
%https://www.alanshawn.com/latex3-tutorial/#macro-expansion-control-v
%
%   Constructing a command in token list
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% #1 to hold the result of the simplication
% #2 the expression
\tl_set:Nx #1 { #2 }
\regex_replace_all:nnN { +- } { - } #1
\regex_replace_all:nnN { -- } { + } #1
%\regex_replace_all:nnN { 1x } { x } #1 <-- wrong (for instance 11x)
\regex_replace_all:nnN { (^|[^\d])1([a-zA-Z\c{[A-Za-z]*}]+) } { \1\2 } #1
}

\ExplSyntaxOff
\begin{document}
\simp{\mysimp}{5+1x}
$\mysimp$


\simp{\mysimp}{5+11x}
$\mysimp$

\simp{\mysimp}{5+1\cos(t)}
$\mysimp$

\simp{\mysimp}{1\ln(x)}
$\mysimp$

\simp{\mysimp}{11\ln(t)}
$\mysimp$

%\newcommand{\a}{1}<----  ERROR
\newcommand{\mya}{1}
\newcommand{\myb}{-1}
%
% i use the name \mysimp, you can use another name
%
\simp{\mysimp}{\mya+\myb}
$\mysimp$


\renewcommand{\mya}{2}
\renewcommand{\myb}{-2}
\newcommand{\myc}{1}
\newcommand{\myd}{-3}
\simp{\mysimp}{\mya+\myb}
\simp{\mysimpdeux}{\myc x-\myd}
$\frac{\mysimp}{\mysimpdeux}$

\renewcommand{\mya}{-3}
\renewcommand{\myb}{3}
\renewcommand{\myc}{-1}
\renewcommand{\myd}{5}
\simp{\mysimp}{\frac{\mya+\myb}{\myc \cos(t)-\myd}}
$\mysimp$
\end{document}

EDIT4 A beginning of answer for the simplification of fraction.

            \documentclass{article}
        \ExplSyntaxOn
        \begin{document}
        \tl_set:Nn \l_tmpa_tl { -\ln(x) }
        $\l_tmpa_tl \rightarrow$           
        \regex_replace_all:nnN { \- \c{ln} } { \c{log} } \l_tmpa_tl        
        $\l_tmpa_tl$\\
        %
        \tl_set:Nn \l_tmpa_tl { -\frac{-3}{4} }
        %\tl_analysis_show:N \l_tmpa_tl 
        $\l_tmpa_tl \rightarrow$           
        \regex_replace_all:nnN { \- \c{frac} \cB\{ \- 3 \cE\} }  { \c{frac} \cB\{ 3 \cE\} } \l_tmpa_tl        
        $\l_tmpa_tl$
        \ExplSyntaxOff
    \end{document}

Answer 2

This is an absolutely tedious problem if you stick to LaTeX. If you're willing to use an external program it becomes much easier. LaTeX is an amazing document preparation system. Using the sagetex package will give you access to a computer algebra system, called Sage and the Python programming language. Here's a quick solution using sagetex.

\documentclass{article}
\usepackage{sagetex}
\begin{document}
\begin{sagesilent}
a = 15
b= -17
c = 1 
d = -1
def Add(a,b):
  if b>=0:
    return r"%s+%s"%(a,b)
  else:
    return r"%s-%s"%(a,abs(b))
def Sub(a,b):
  if b>=0:
    return r"%s-%s"%(a,b)
  else:
    return r"%s+%s"%(a,abs(b))
def Poly(a,b):
  R.<x>=ZZ[] 
  return ax+b
\end{sagesilent}
Calculating $a+b$ where $a = \sage{a}$ and $b = \sage{b}$ 
yields $\sagestr{Add(a,b)}=\sage{a+b}$. Moreover $a-b$ is 
$\sagestr{Sub(a,b)}=\sage{a-b}$. For the linear expression $ax+b$ where
$a=\sage{d}$ and $b=\sage{b}$ we have $\sage{Poly(d,b)}$. Reversing the
terms gives $\sage{Poly(b,d)}$. Sage can even handle the multiplication of 
polynomials: $(\sage{Poly(c,d)})\cdot(\sage{Poly(b,b)})=\sage{Poly(c,d)Poly(b,b)}$.
\end{document}

The result running in Cocalc is below:

The scratchwork (setup) where you assign variables and define your functions is in the sagesilent blocks. This does not show in your document. Sage output used here involves \sage and \sagestr; \sage is for calculations (i.e. things in math mode) while \sagestr is for strings. To avoid problems with various LaTeX characters, such as \ we use r"" (raw string). The function Add(a,b) can be reasoned as follows: if b>=0 then return the raw string r"%s+%s"%(a,b). The %s is a placeholder for string data. What goes into the 2 spots are a and b. Since this string involves numbers, I call it as $\sagestr{Add(a,b)}=\sage{a+b}$. Sage, being a CAS, gives you the answer of a+b as a number when \sage{a+b} is called. If b is negative then return r"%s-%s"%(a,abs(b)) returns the raw string a minus absolute value of b, avoiding the +- problem. With polynomials, the power of Sage starts to shine. The function Poly(a,b) starts by declaring that we are working in the ring of polynomials with integer coefficients, R.<x>=ZZ[]. Tell Sage what a and b are and it takes care of reconciling your +- problems. Moreover, by working with a mathematical object as opposed to a string I can then have Sage take care of multiplying the polynomials together with \sage{Poly(c,d)*Poly(b,b)}.

Search this site for more sagetex examples. I've answered questions involving polynomials here and here.

Finally, Sage is not part of your LaTeX distribution. You can sign up for a free Cocalc account to experiment with sagetex and Sage. If you like it, it's possible to download Sage to your computer and sync it with your LaTeX distribution. This has been problematic for some; Cocalc is the easiest way to get started.