As Reggie nods in approval.

WASHINGTON — President Barack Obama felt a sense of relief after coming out in favor of gay marriage, according to “Double Down,” a new book about the 2012 campaign.

“I feel so good about that,” he reportedly told adviser David Plouffe, after declaring his support in an interview with ABC’s Robin Roberts.

But the Obama team didn’t feel “so good” several days before, when Vice President Joe Biden went off script in a “Meet The Press” interview, expressing his support for gay marriage and getting out in front of the president.

The anecdote is one of many in the richly reported book, set to be released Tuesday, by “Game Change” authors Mark Halperin and John Heilemann. It also details how Obama advisers considered replacing Biden with then-Secretary of State Hillary Clinton on the 2012 ticket and how Jon Huntsman, Sr. was the source for Senate Majority Leader Harry Reid’s claim that Mitt Romney hadn’t paid taxes for 10 years. The Huffington Post independently obtained a copy of the book before its release.

Biden, whom the authors describe as someone with a “nonfunctional” gaydar who didn’t know how many gay staffers he had, experienced an epiphany at the end of April, after appearing at an LGBT fundraising event at the home of a gay couple with two kids in Los Angeles.