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MELBOURNE — Roger Federer was not going to go gently, of course, no matter how daunting the number of match points — his opponent accumulated seven! — no matter how achy his 38-year-old legs, no matter how slow his serves, no matter how off-target his groundstrokes.

Federer still plays for the love of these stages and circumstances. Still yearns for more trophies, too. Down to his very last gasp, time and again, against someone a decade younger, 100th-ranked Tennys Sandgren of the United States, Federer somehow pulled off a memorable comeback to reach the Australian Open semifinals for the 15th time.

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Despite all sorts of signs he was not quite himself for much of the match, Federer beat the biceps-baring, hard-hitting, court-covering Sandgren, 6-3, 2-6, 2-6, 7-6 (10-8), 6-3, on Tuesday in a rollicking quarterfinal that appeared to be over long before it truly was.

“For the most time there, I thought that was it. Of course, there’s little sparkles where maybe not. Then you’re like, ‘No, it is over,’ ” said Federer, who only once before had won after facing as many as seven match points, equaling his personal best from all the way back in 2003. “Only maybe when I won that fourth set did I really think that, maybe, this whole thing could turn around.”

He said afterward that it had been his groin muscle that was the problem and he couldn’t be certain whether he would be fully recovered for his next match. That will come against defending champion Novak Djokovic, who overwhelmed No. 32 Milos Raonic, 6-4, 6-3, 7-6 (7-1), to improve to 10-0 against the 2016 Wimbledon runner-up.

“He was just too good,” Raonic said.

In the women’s draw Wednesday, No. 4 Simona Halep gained a berth in the semifinals with a dominating 6-1, 6-1 win over No. 28 Anett Kontaveit. Halep will face unseeded Garbine Muguruza, who eliminated No. 30 Anastasia Pavlyuchenkova, 7-5, 6-3, in the day’s other quarterfinal.

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Top-seeded Ashleigh Barty will face American Sofia Kenin in the other women’s semifinal.

The Federer-Djokovic meeting will be the 50th of their careers; No. 3 Federer has won 20 Grand Slam titles, and No. 2 Djokovic owns 16.

Djokovic leads their head-to-head series, 26-23, including their past five matches at majors.

“Roger is Roger. You know that he’s always going to play on such a high level, regardless of the surface,” Djokovic said. “He loves to play these kind of matches, big rivalries, semis, finals of Grand Slams.”

About the only thing that slowed Djokovic’s progression to a 37th career Grand Slam semifinal — Federer earned his 46th — was the medical timeout the Serb asked for at 4-all in the third set so he could put in new contact lenses.

“It was just something I had to do,” Djokovic said, “because those few games, I really couldn’t see much.”

Alexander Zverev, a 22-year-old from Germany, reached his first major semifinal by overcoming a terrible start Wednesday at Melbourne Park and putting together a 1-6, 6-3, 6-4, 6-2 victory over three-time Grand Slam champion Stan Wawrinka.

“I hope,” Zverev said, “this will be the first of many.”

After ceding the opening set in 24 minutes, Zverev regrouped and recalibrated his strategy, using all of his 6-foot-6-inch frame to get to balls along the baseline and stretch points until Wawrinka faltered.

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Zverev’s sometimes-shaky serve was suddenly terrific, and Wawrinka’s barrel-chested baseline bashing weakened, as if he might be injured. Wawrinka finished with five winners and 31 errors on that side, 18 unforced and 13 forced.

On Friday, Zverev will take on No. 1 Rafael Nadal or No. 5 Dominic Thiem for a berth in the final. Nadal vs. Thiem was scheduled for Wednesday night local time.