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I agree that 4.Nf3 is the most accurate move against the Budapest, and leads to a significant White advantage.
From my database results 4. Bf4 is played more and scores better. Is there something significantly different between 4.Nf3 and 4.Bf4 that I'm missing?
And I actually feel as though black might actually have some good chances after this line:
Blacks bishop is in trouble there. White will develop faster after the black knight on g4 takes the Epawn. Be2 Whites Knight will threaten and trade or take the black bishop as the bishop retreats or has to go to b4 and then capture on d2.
White will develop faster and have both bishops with tons of space. An easy win for white in theory.
Basically it leads to a +/- position in the independent line (4...Bc5) and otherwise is the same as the 4.Bf4 line (to which it often transposes if Black does not play 4...Bc5). So it's theoretically more accurate.