Just like Federer, Barty was still searching for stability after two matches in New York. And just like Federer, she may have found it by relying on a singular shot, perhaps the most overlooked, underrated and stealthy shot in all of tennis: the backhand slice.

Hoping to regain a measure of consistency, Federer unfurled his backhand slice early against Dan Evans on Friday. Instead of playing loose, instead of rolling out his powerful topspin in the first set, he let backspin add a needed measure of conservative play to his game.

He deployed it often on returns, during rallies and while coming to the net — throwing in a constellation of slow spinners that baited Evans into match-altering errors. The end result was vintage Federer: a 6-2, 6-2, 6-1 win. Clean, efficient, confident, and the kind of game the Swiss will need to get by a tougher opponent, 15th-seeded David Goffin, in Sunday’s fourth-round match.

The undercut backhand provided a similar safe space for Barty, who is known as having the best slice among the women. On Friday she did not hit her signature shot with quite the clarity she has when in peak form, but it was the foundation that helped her move past Maria Sakkari, 7-5, 6-3, in what was clearly her best performance of the tournament.

When both matches were done, fans could be heard rhapsodizing about the awe-inspiring sweetness of the Federer and Barty serve and forehand, the glory shots that draw all the attention. That shouldn’t come as a surprise. Asked after his match to name the strokes that give him the most pleasure, Federer didn’t hesitate before citing those two remarkably forceful shots. So often when he needs to rely on them, he said, “the rally is over.”