It's impossible to give a foolproof formula that will work for everyone, because people have different family situations and income sources and tax credit eligibility that can affect their taxes. However if we keep it pretty simple, assume you're an early retiree with no income other than from taxable index funds and Roth conversions, there are a few numbers you need to know.
First: the standard deduction. In 2017, this amount is $12,700 for married couples, $9,350 for head of household, and $6,350 for singles. If you have itemizable deductions in a higher amount than this, you can deduct more.
Next: the personal exemption. This is $4,050 for yourself, your spouse, and each dependent.
The sum of your deductions and exemptions is the amount of tax-free regular income (such as wages, interest, non-qualified dividends, Roth conversions) you can have. A married couple with two kids at home who doesn't itemize would then get $12,700 + ($4,050 x 4) = $28,900 of any type of income free of federal tax, while a single person with no dependents would get $6,350 + $4,050 = $10,400.
On top of this, qualified dividends and long-term gains are tax-free up to the top of the 15% tax bracket. This amount is $37,950 for singles, $50,800 for heads of household, and $75,900 for married joint filers.
Other things like the child tax credit and the foreign tax credit can increase your tax-free income space a bit more.
However you should be aware that if you're buying health insurance through the Obamacare exchanges (while they last), this tax-free income still counts for determining your health insurance subsidies. This means that although the nominal tax rate for this income is 0%, you'll gradually lose your health insurance subsidies by a few cents for each dollar of income.