When is it better to underpromote?

u mean promoting 2a 'stretched' knight throwing dn a forky worky ?

If by stretched knight, you mean camel then yes.  A chess camel moves 3 squares horizontally, and one vertically, or 3 squares vertically and one horizontally, and so would fork the king and queen.

you sometimes have to underpromote to avoid stalemate, like in the saavedra position:

white to move can only win by promoting to a rook

but why would a competent engine fall for such trick?  Surely a competent engine  would realize that the better move would be to take the underpromoted piece.   Or am I misunderstanding what you are talking about?  Perhaps you could post a game demonstration of what you are referring to?

An engine starts by making all available one moves on the board and evaluating each move. From that list, it finds the top 5 or 10 or X most interesting moves(often called candidate moves) and does longer calculations to chose its move. 

An engine will fully calculate their promotion to the maximum, but the engine may under promote to attempt to slide under what its opponent considers a candidate move. 

Taking a queen would always be a candidate move by its opponent, but maybe not taking a knight or a bishop. Overall, that means the engine is "up" because it kept its minor piece, but would have lost its major piece. The engine also has a use for its minor piece, since it has already calculated a plan, so it is way ahead of the game if its opponent missed it.

If I understand correctly, the under promotion is done to in an attempt to trick the opponent into not  taking the underpromoted piece.  Therefore the player making the underpromotion  must think he/he/it is in a better position if the opponent does not take the under promoted piece.   If he/she/it is correct and they are in a better position if the underpromoted piece is not taken, why wouldn't competent modern chess engine also realize this and therefore take the underpromoted piece?  Again perhaps someone could make a demonstration and this might clear things up.  

Competent modern chess engines do not "realize". That is what humans do. That chess engines do is run weighted calculations and change values. A promoted piece could be a "poisoned" piece, where taking it causes a loss of the game or just material. By ranking a promoted piece as too valuable, then you run the risk of accepting the poisoned piece and losing the game. By forcing an engine to always evaluate promoted pieces, you may cause it to spend too much of its energy on a dead end.

There's a balancing act here.

Here's an example of Leela under promoting to keep a knight. 

https://youtu.be/hoi_QnOdqj0?t=1361

You know what I mean.

Unless they’ve found a way to give chess engines emotions, it wasn’t trolling.  I don’t know why it didn’t checkmate once it had a queen, but it wasn’t trolling. 

I could call it trolling; come on @Warbler you have to see the humor in this. I agree with IM Daniel Rensch: "This is not what computers do to win." 

Is it possible that it did all because was less moves to mate?  Is it possible it could be a bug in the programming?

Leela is a learning program, rather than an algorithm like Stockfish or Komodo. The problem with learning programs is that we can't say for certain why it does what it does, where as we can say with good confidence why stockfish may do something. 

Leela often misses strong attacks that even humans can see, because it seems to value taking away opportunities from its opponent and simplifying the position in her favor higher than actually winning the game. This is not necessarily a "bug", since this emphasis is what makes Leela a top rated computer opponent with some insanely strong midgames compared to other engines, but it also makes Leela do awkward things in the end game, since Leela seems to play a midgame all the way until the mate. 

The least amout of pieces needed for a knight promotion on each side with White promoting first.

Why wouldn’t white just promote to queen?

1.e8Q+ Kg7 2.Qf7+ Kh6 3.Qh7+ Kg5 and still several moves to checkmate.

1.e8Q+ Kg7 2.Nh5+Kh6 3.g7 and still several moves to checkmate.

Underpromotion leads to faster checkmate, and therefore is correct.

Coming off from #61, here are the minimal(?) setups for mutual Rook and Bishop promotion with White promoting first.

Why not post #66 as puzzles, RewanDemontay? They are nice problems!

did you know the babson task ?

Yes, I do know of it.

In Russian, all known orthodox problems, including cyclic Babsons: http://superproblem.ru/archive/P/Babson.html

(It has more examples then Tim Krabbe’s collection.)