simplifying formulas in maxima

Question

I'm using maxima software to help me simplify formulas. Usually, I can manage easily with subst, ratsubst, factor, ratsimp, ... But there are still are few formulas I find hard to simplify the way I'd like to.

1. assuming that a > b and c > d, I'd like to simplify fractions starting with a - sign this way :

   -(a - b)/(d - c)    ->   ( a - b )/( c - d)
   

   but I don't how to do it. It seems that maxima simplifier algorithm will try to sort variables in its own way.

   I created my own maxima function to try to simplify these useless minus signs.

   no_minus(fraction):=
     block([simp:true,
        numerat:expand(-ratnumer(fraction)),
        denominat:expand(-ratdenom(fraction))],
       block([simp:false],
         numerat/denominat));
   -a/(b-x);
   no_minus(-a/(b-x));
   no_minus(-a*b*c/(b-x*b*f-f));
   

I'd expected no_minus(-a/(b-x)) would have returned a/(x-b) but it didn't.

2. I'd like to introduce a new infix operator to denote that two expressions are approximately equal. For instance, if x is approximately equal to y. I'd like to note it

   x =~ y
   
   infix("=~").
   

How to configure the simplifier so that when the input is

2*x+3 =~ u+v;  
(%-3)/2;

the output is

x =~ (u+v-3)/2

Answer 1

Well, you can define simplification rules via tellsimp (and tellsimpafter, defrule, and defmatch). Maybe this is enough to get started.

(%i3) infix ("=~") $
(%i4) matchdeclare ([aa, bb, cc], all) $
(%i5) tellsimp ((aa =~ bb) * cc, (aa * cc) =~ (bb * cc)) $
tellsimp: warning: rule will treat '?mtimes' as noncommutative and nonassociative.
(%i6) tellsimp ((aa =~ bb) + cc, (aa + cc) =~ (bb + cc)) $
tellsimp: warning: rule will treat '?mplus' as noncommutative and nonassociative.
(%i7) (2*x + 3) =~ (u + v);
(%o7) (2*x+3) =~ (v+u)
(%i8) (% - 3)/2;
(%o8) x =~ ((v+u-3)/2)