@Christopher2222 I'm sorry, but I cannot understand what you're now asking because it seems as if you haven't understood a single word I've written so far. I have to stop with this now. Good luck.

@Christopher2222 But generally computed results will not show the effect. Eg,

restart;
macro(v^2=a^2+b^2):

p:=v:
p*v;
                  2
                 v 

@radaar How could I know anything about a .dll file that you merely reference in a worksheet!?

Is it for 32bit or 64bit?  (Your attached worksheet was last saved in 64bit Maple 2017.3. So if you're using 64bit Maple then the .dll better match.)

What's inside it? How did you build it?

@ecterrab Perhaps I misunderstood, but it seems to me that the OP was looking for considerably more than what macro provides.

Consider these examples. I got a strong sense the the OP wanted Maple to be able to utilize the "assignment" for mathematical effect on subsequently formed expressions -- rather than merely an abbreviation mechanism for input.

restart;
macro(v^2=a^2+b^2):

foo := (v^2)^2;

                             4
                     foo := v

hmm1 := foo/(a^2+b^2);

                               4
                              v
                    hmm1 := -------
                             2    2
                            a  + b

simplify(hmm1);

                         4
                        v
                      -------
                       2    2
                      a  + b

hmm2 := hmm1/v^2;

                              2
                             v
                   hmm2 := -------
                            2    2
                           a  + b

simplify(hmm2);

                          2
                         v
                       -------
                        2    2
                       a  + b

hmm3 := hmm2*v^2;

                               2
                      hmm3 := v

simplify(hmm3);

                          2
                         v

Please don't repost the question just because you haven't received an answer.

@Joe Riel Yes, there some special assignment syntax cases that have a specific meaning and are useful. But his meaning seems to be either general or at best unclearly specified.

Perhaps he is talking about assigning into rember tables of procedures. It doesn't seem clear though, because he also seems to be talking about subsequent use in contexts where ambiguity comes into play. I find it hard to discern what precisely he's requesting, that wouldn't be better achieved via algsubs, say.

I'm not saying that there is no way to extend Maple's assignment functionality in ways that could ever be useful. I'm saying that, as described so far, the request seems unclear or ambiguous.

@Christopher2222 No, that is not a good point. It muddles concepts and doesn't make sense in general.

And in fact you have your example wrong/backwards. But even if you corrected it then you'd be in a position that I anticipated when I wrote above that instead you should utilize an equation. That's because in general your proposal is unworkable, generally ambiguous, and does not make sense.

It doesn't make sense, as you've described it.

If you want to utilize an equation... then do so. You can even assign the equation (which you may utilize) to a name.

But what you've described does not make sense that I can see. You are muddling concepts.

@gaurav_rs Using Maple 2017.2,

for dig from 10 to 20 do
  Digits:=dig;
  forget(evalf); forget(int);
  res := int(fnormal(F1), [theta= Pi/4..7*Pi/4, r= 0..1]);
  printf("Digits=%a  %f\n",dig,res);
end do:

  Digits=10  -1152122349000.000000
  Digits=11  -1116001007100.000000
  Digits=12  -1116001007190.000000
  Digits=13  -1116001007191.000000
  Digits=14  -1116001007191.800000
  Digits=15  -1116001007191.800000
  Digits=16  -1116001007191.796000
  Digits=17  -1116001007191.796500
  Digits=18  -1116001007191.796470
  Digits=19  -1116001007191.796470
  Digits=20  -1116001007191.796470

@Carl Love Yes, of course, thanks.

@Kitonum In my first Reply/Comment on the Question I mentioned, "It might also matter whether you need to be able to export them (say to an animated GIF file) which plays them together."

I wrote that because the method that you've shown right-click exports only one of the animations to a single animated GIF file.

That related to why I wrote an Answer as I did.

@Kitonum It'll be interesting to find out what the OP really wanted. I interpreted "...sync two or more different animations" to mean two or more separately displayed animations and have them play in a synchronized manner.

In a followup Comment he added, "...just different x and y scale which is irrelevant since they will be on different plots."

I figured the OP meant in unison and separately when he wrote sync and different. Perhaps you are right, though, in interpreting that he meant together.

@Stretto And did you read the post to which I linked? It seems to cover your example.

In case it may relate to your goal, you could look at this stackoverflow post. Note that it deals with a synchronized display of an array plot, rather than two animations which have already been constructed.

One aspect that seems relevant is whether the two animations have the same number of frames (one a multiple of the other). If that's not the case then it could get tricky -- possibly requiring an adjustment if you want them to finish/loop concurrently.

It might also matter whether you need to be able to export them (say to an animated GIF file) which plays them together.

Perhaps you could supply your actual example, with full detail to reproduce.

What does the sum (of numeric integrations) over the xx[i] represent?

Is it possible that you are trying to integrate (the inner numeric integrals) over "xx" in a given range? I mean, does the final sum total, divided by the number of terms in the sum, (possibly in the limit) represent something meaningful in your goal?

If so then it might be a better idea to simply numerically integrate over one more dimension.

Just an thought. I have little idea of what you're really trying to accomplish because you haven't explained it.