If you've already inputted information, but not yet filed, do the errors get corrected/recalculated automatically? Or should one wait and enter information when all known bugs are fixed?
If they're doing their job right and you've entered your data accurately, you could enter your data now and wait for the fixes later and *probably* be OK.
The one exception that comes to mind is if you're using some sort of import function to pull in, say a consolidated 1099 or a W-2. If there is an issue with their import software, I suppose it's possible that could affect the import of your data. If that happened, you'd probably have to choose between deleting the bad import and starting over, or manually correcting the data that was imported.
The other possible error would be if their interview code is broken; it's possible you answer a question correctly but their code leads you down a wrong path and you never end up putting in the data because of it. In this case the main way to fix would be to go through the interview process again and hope they steer you the right way.
In theory, the commercial tax software programs could keep track of bugs and bug fixes and let you know when they apply to your return. But in practice, this is actually more complicated and more work and itself susceptible to error, so I think they don't try to do so. They probably just publish release notes and tell you to update your software before filing.
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I file my and my entire family's returns through AARP Foundation Tax Aide where I volunteer. They use TaxSlayer, which has bugs too, but I get to see the bug lists and can generally tell which apply and which don't. I also get to see when they're fixed, and I can look at the return itself and usually can tell if there is a problem. For regular customers, Tax Aide also always does a second full quality review by a second tax preparer. The quality review sometimes catch errors, so the overall program accuracy rate is very very high.
If you're doing your own return, you can sort of replicate a quality review by doing your return one day, then coming back a week later with fresh eyes and looking it over again before filing. Even if you hire a professional tax preparer, I would recommend reviewing their work. They're usually competent, but some aren't, and the competent ones are busy enough to where they can make errors just due to sheer volume and the time pressure that comes with the desire to be profitable.