How can one precisely sacrifice a piece for a winning attack? r2qkb1r/5ppp/p1bppn2/1p3P2/4P3/1BN5/PPP3PP/R1BQ1RK1 b qk - 0 1

1... b4 2. fxe6! bxc3 3. exf7+


In the above game, black played 1...b4 to encumber the white knight's journey to d5. Ambitiously, white sacrificed it with 2. fxe6! bxc3 3. exf7+: attacking the uncastled king, claiming the f7 square as his, and trapping the king on the center files. 

The inactivation of black's dark bishop justifies this sacrifice: it's not soon seeing daylight anytime. Similarly, the a8 rook is also passive; only the queen, light-squared bishop, and f6 knight appear to be able to defend a central onslaught. 

The black king can move to either Kd7 or Ke7, which aren't ideal in the least: these squares lie on two semi-open files available for white to take advantage of.

Altogether, this convinces white that black's life is going to be a terrible time. Still, the development of the queen's bishop must occur for placement of the connected rooks on d1, e1. He can choose squares such as g5, pinning the queen, or move his bishop along the a3-f8 diagonal after a capture on c3.


White needn't checkmate, but he should return with, beyond question, a better position than black. What final position does white visualize that warrants the knight's goodbye?
There can be an overwhelming number of variations in realizing that final position: after 4. Re1, black has a number of responses. If this is the best move, how does white calmly reassure himself that all of its responses are inadequate? If it's not the best move, what is and why?


I am not looking for a final, deeply analyzed (e.g. best) variation, but a winning position and the most effective way to achieve it: the moves that should be played first, and why. A 3-ply variation is enough, but a 4-ply variation is perfect. Maybe I also missed some critical details about the board, and would welcome these observations, if any.
If this is the best move, how does white calmly reassure himself that all of its responses are inadequate?


White doesn't see a forced advantage in every line. He sees, what he believes to be, good winning chances. To quote Spielmann in The Art of Sacrifice in Chess:


  In real sacrifices the player gives up material but is unable to calculate the consequences with accuracy; he has to rely on intuition





  What final position does white visualize that warrants the knight's goodbye?


I think you answered your own question. After 2. fxe6 bxc3 3. exf7+ white probably thought that a lead in development and 2 pawns for the piece (potentially 3 if bxc3 is played) is sufficient compensation.

White doesn't have to look further than this if they are willing to take risks and figure out the attack 2 or 3 moves at a time. If white is risk averse, then they may not even consider 2. fxe6 bxc3 3. exf7+ at all.