Let’s not confuse the law with company policy. The law (U.S. Federal) is that any income you receive from an employer is taxable – whether base salary or bonus. There is also social security tax (amusingly called “contributions”) up to the ceiling and Medicare tax, paid by both the employee and employer.
The employer must (under most conditions) withhold income tax and social security tax at the time payment is made, whether base salary or bonus.
Employees fill out forms that list the number of dependents, so that employees can choose to have more or less income tax withheld (those forms have no impact on social security contributions.) The employer has very little flexibility in terms of amounts to be withheld, once the employee has made such choices.
OK, that’s tax law.
Now. Companies can choose to pay on either a gross (before tax) or net (after tax) basis. You’ve focused on bonus, but why stop there? How about base salary – the company tells someone that their salary is $30,000. Does that mean they have to pay taxes and so only receive (say) $20,000? or Does that mean that they really get (say) $45,000 and pay taxes so that they have $30,000 left over?
Most companies describe salary and bonus on a gross basis – that is, they describe the amount they are going to pay the employee. That amount is taxable income, and the employees keep whatever is left over after tax. If you are told your salary is $30,000 and your bonus is $3,000 then you will report a total of $33,000 on your income tax form and you may pay (let’s estimate) $11,000 in taxes, so you keep $22,000.
It is very rare for an employer to tell that employee that they will receive a net salary of $22,000. For one thing, it’s a smaller amount, so the company would rather you think of your salary as higher. For another thing, the company doesn’t want to get involved in individual circumstances – “Hey, I have six dependent children and a big mortgage, so my taxes are less than Fred’s, who is single and rents…”
The main situation where an employer commonly communicates a net salary is when a U.S. citizen is being transferred outside the U.S. – very often the company promises a net salary amount, because the foreign taxes may be much higher than U.S. taxes would be, so the communication of a guaranteed net salary is attractive for the person being transferred.
I have never heard of a situation such as described above, where the company said the employees would get a $200 bonus and then said they meant “net” (after tax) rather than “gros” (taxable or before tax.) Sounds like someone made a bonehead move. You should try complaining about your base pay on the same basis – that you were told you’d get $X salary, but you have to pay taxes, so you should get more money so that you hve $X left after taxes. Same principle. See what they say.
On the math – not sure if you’re asking about the math behind how they figure the amount to pay. It’s pretty easy. You want someone to receive $200. You check that the average tax rate for their income level is 20% (say). Then your formula is:
200 / (1 - .20) = $250
Net pay / (1-tax rate) = Gross Pay