user-1: I've had a few looks at the Traxler counterattack before. It seems fine for Black, but somehow the engine (Stockfish) says that White has a +2.57 advantage after Nxf7, but not much advantage after Bxf7+. Why is this so, when this opening is accepted by masters?

user15223: I can't find it now, but I remember reading this exchange on Chessbase. A player just played ...e6 on one move, and then ...e5 on the next, seemingly losing a tempo for no reason. It was clearly computer preparation however. An interviewer asked GM Viswanathan Anand what he thought about the moves. Anand replied that if the computer thinks it's fine, then it's fine. Instead you have to analyze the lines afterwards to see why it's fine.
In other words: these days computers are so strong that their judgement trumps human judgement, including those made by world champion-caliber players.
Now the fact is that computers don't like the Traxler, even at high depth. It was played in the TCEC S17 superfinal between Leela and Stockfish. Here's the first game, and here's the reverse. Note:

These engines are playing on super hardware and achieve speeds and depths that would take very long for ordinary machines. When Leela played her fifth move, she was at depth 22/62. When Stockfish played its fifth move, it was at depth 43/62.
Both finalists thought 5.Bxf7+ is the best move.
Both finalists evaluated the position after 5.Bxf7+ as +1.5 for White (more precisely, Stockfish gave +1.5 for White; Leela thought White had a 60% chance to win).
Comparatively, in the starting position, White has roughly a +0.5 advantage.
There is some fluctuation in engine eval, so e.g. if Stockfish thinks a move that is +0.1 pawns better than the next-best move, the next-best move might still be playable. However, +1 is a pretty large gap.

Therefore the only conclusion here is that the human masters are wrong. The Traxler is objectively dubious at best. Playing this can work against humans because humans cannot find a string of "only moves" in a row to hold a position, plus it's psychologically difficult to defend against a strong attack for a long time. That's why humans - even top-level humans- often crack under pressure. The Traxler also offers plenty of opportunities for both sides, which is great if you want to play for a win and/or you think you are better than your opponent. But if you play the Traxler as Black in correspondence chess, you are asking for trouble. Maybe 32-piece tablebases will show that your position isn't lost, but you would have to fight hard for equality.
Finally I'll add that it's misleading to say the Traxler is well-regarded by masters:

Wikipedia's article on the Two Knights includes this quote: "No grandmasters have regularly adopted the Wilkes-Barre as Black, but Alexander Beliavsky and Alexei Shirov have played it occasionally even in top competition"
Chess Openings Explorer gives by far the most common response to 4.Ng5 as 4...d5 (which is also the main line of the Two Knights).

I'd call the Traxler playable in human games, but only as an occasional weapon.

-1: <p>I've had a few looks at the Traxler counterattack before. It seems fine for Black, but somehow the engine (Stockfish) says that White has a +2.57 advantage after Nxf7, but not much advantage after Bxf7+. Why is this so, when this opening is accepted by masters?</p>
 15223: <p>I can't find it now, but I remember reading this exchange on Chessbase. A player just played ...e6 on one move, and then ...e5 on the next, seemingly losing a tempo for no reason. It was clearly computer preparation however. An interviewer asked GM Viswanathan Anand what he thought about the moves. Anand replied that if the computer thinks it's fine, then it's fine. Instead you have to analyze the lines afterwards to see why it's fine.</p>
<p>In other words: these days computers are so strong that their judgement trumps human judgement, including those made by world champion-caliber players.</p>
<p>Now the fact is that computers don't like the Traxler, even at high depth. It was played in the TCEC S17 superfinal between Leela and Stockfish. <a href="https://tcec-chess.com/#div=sf&amp;game=91&amp;season=17" rel="nofollow noreferrer">Here's the first game</a>, and <a href="https://tcec-chess.com/#div=sf&amp;game=92&amp;season=17" rel="nofollow noreferrer">here's the reverse</a>. Note:</p>
<ul>
<li>These engines are playing on super hardware and achieve speeds and depths that would take very long for ordinary machines. When Leela played her fifth move, she was at depth 22/62. When Stockfish played its fifth move, it was at depth 43/62.</li>
<li>Both finalists thought 5.Bxf7+ is the best move.</li>
<li>Both finalists evaluated the position after 5.Bxf7+ as +1.5 for White (more precisely, Stockfish gave +1.5 for White; Leela thought White had a 60% chance to win).</li>
<li>Comparatively, in the starting position, White has roughly a +0.5 advantage.</li>
<li>There is some fluctuation in engine eval, so e.g. if Stockfish thinks a move that is +0.1 pawns better than the next-best move, the next-best move might still be playable. However, +1 is a pretty large gap.</li>
</ul>
<p>Therefore the only conclusion here is that the human masters are wrong. The Traxler is objectively dubious at best. Playing this <em>can</em> work against humans because humans cannot find a string of &quot;only moves&quot; in a row to hold a position, plus it's psychologically difficult to defend against a strong attack for a long time. That's why humans - even top-level humans- often crack under pressure. The Traxler also offers plenty of opportunities for both sides, which is great if you want to play for a win and/or you think you are better than your opponent. But if you play the Traxler as Black in correspondence chess, you are asking for trouble. Maybe 32-piece tablebases will show that your position isn't lost, but you would have to fight hard for equality.</p>
<p>Finally I'll add that it's misleading to say the Traxler is well-regarded by masters:</p>
<ul>
<li><a href="https://en.wikipedia.org/wiki/Two_Knights_Defense#Traxler_Variation:_4...Bc5" rel="nofollow noreferrer">Wikipedia's article</a> on the Two Knights includes this quote: &quot;No grandmasters have regularly adopted the Wilkes-Barre as Black, but Alexander Beliavsky and Alexei Shirov have played it occasionally even in top competition&quot;</li>
<li><a href="https://www.chessgames.com/perl/explorer?node=144658&amp;move=4.5&amp;moves=e4.e5.Nf3.Nc6.Bc4.Nf6.Ng5&amp;nodes=21720.21721.21722.21723.24145.24146.144658" rel="nofollow noreferrer">Chess Openings Explorer</a> gives by far the most common response to 4.Ng5 as 4...d5 (which is also the main line of the Two Knights).</li>
</ul>
<p>I'd call the Traxler playable in human games, but only as an occasional weapon.</p>
