Here are the numbers. Musk exercised 5,503,972 stock options with a net gain of $208.70 per share, valued at a sum of $1,148,678,956.40. To cover the taxes on that sizable amount, he sold 2,782,670 shares worth about $593 million. As Electrek notes, it is "weird" for a billionaire to pay almost $600 million in taxes, but it's in keeping with Musk's attitude about stocks. Electrek points to an email that was published in Ashlee Vance's biography about Musk in which Musk explained to SpaceX employees how to sell their stock with an eye to tax rules:
If you'd like to get a guided tour of Musk's SEC filings, read this detailed post on Teslarati. Electrek also says that, given the ways the numbers line up and past history, "it's not difficult to believe Tesla's claim that all proceeds from Musk's stock sale will go toward paying taxes." Unless you're a TSLA critic, of course.
To maximize your post tax return, you are probably best off exercising your options to convert them to stock (if you can afford to do this) and then holding the stock for a year before selling it at our roughly biannual liquidity events. This allows you to pay the capital gains tax rate, instead of the income tax rate.
You can read Elon Musk's Form 4 SEC Filing for yourself here.