Prior to reading this: I notice your math uses 12% as his call3bet% and 88% as his foldto3bet%. The problem is that you assign him 12% range, when his range should be 9.3% (.78*.12). This slightly changes the equity if you're called.
You're calculating the EV of the decision of shoving ATC
given that your opponent minraised.
This means that p(opens) shouldn't show up in your equation (you're examining the game for after he opens, not before it). When dealing with conditional probabilities, you're treating them as if the associated probability is equal to 1 or 0 (depending on if it meets the condition).
So your equation:
<shoving ATC> = p(opens)[p(folds)(pot size) + p(calls)p(we win)(pot size)-(eff stack size)] + p(open folds)(blinds)
Needs to be rewritten as:
[p(folds)(pot size) +p(calls)[p(we win)(eff stack size)-p(lose)(eff stack size)]
Notice that we dropped the p(open folds) term because we're defining this as "the situation where he minraises," which means p(open folds) = 0, so we ignore this term.
Now, when you plugged in these numbers, it appears you forgot to include therm p(lose), which is 1-.33 = .67. Include this, and you end up with:
(0.88)(3) + (0.12)((0.33)*S - .66*S) = 0
Which is equal to 0 at 66.
So, when your opponent minraise / calls with this ridiculous ratio, you can 3bet jam ATC up to 66 BB deep.
So don't minraise 78% and call 12%