Defeat for Halep means that Karolina Pliskova, at number seven, is the only women's top 10 seed to reach the fourth round. The only past champion at the grass-court tournament left is Serena Williams.
Fourteen of the women's and 16 of the men's seeds have failed to make it through to the second week of the third grand slam of the year.
Last year's French Open champion Jelena Ostapenko eased past the challenge of Vitalia Diatchenko, who had defeated Maria Sharapova in the first round, with a 6-0 6-4 victory. "Because every day, something odd is happening in the draw".
The Spaniard's 6-1 6-2 6-4 victory means Roger Federer can not catch him even if the Swiss goes on to win a ninth title at the All England Club.
In this case, though, Gulbis already has been a major semifinalist - he made it that far at the 2014 French Open, but his ranking slid because of a series of injuries - while the up-and-coming, 21-year-old Zverev has yet to have that sort of breakthrough.
Nadal, with Spain out of the World Cup, was heading to watch England play Sweden.
"It's fantastic. I was 2-5 down in the final set but the crowd pushed me to fight". "She deserved to win".
"I knew she [was] going to fight really hard", Hsieh said. But the 48th-ranked Hsieh got out of that predicament with a backhand victor.
Hosts Russia disappointed but exit World Cup stage with pride
After saving the first, Subasic was aided when Mario Fernandes missed the net with the third attempt. But he saved fantastically the first shot in the shootout. "Thank God everything is fine with him".
Hsieh had shown flashes of ingenuity, but a simple backhand into the net when she had break point to tie the set at 4-4 proved costly as Halep broke again in the ninth game to clinch the opener.
"I was 100 per cent convinced it was a double bounce", Djokovic said. She did a lot of that during her news conference, making jokes and cracking herself up.
Halep won her first Grand Slam at Roland Garros last month, but was scathing of her own performance after seeing her bid for back-to-back majors fall short.
The Belgian, ranked number 47 in the world, knocked out defending champion Garbine Muguruza in round two.
For Hseih, the Taiwanese progresses to the last 16 at Wimbledon for the first time in her career - her previous best was the third round in 2012 - and will next play Slovak Dominika Cibulkova.
Alison Van Uytvanck backed up her impressive victory over Muguruza with an assured 6-2 6-3 success against 28th seed Annett Kontaveit.
Novak Djokovic shrugged off a controversial call against him as the three-time champion hit back to beat Kyle Edmund 4-6, 6-3, 6-2, 6-4 in the Wimbledon third round on Saturday.
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