online softmax

FlashAttention does the same arithmetic as naive attention and is several times faster. The answer is a three-line recurrence and one IO count, and interviewers want both.

The one-query recurrence is a warm-up. A real kernel carries a vector of maxima and a vector of sums for a tile of rows, and the rescale becomes a diagonal matrix. The answer writes it, runs it, and shows the edge case that turns it into NaN.

The tutorial kernel loads a whole row into one block and breaks the moment the row is a 128,256-entry vocabulary. The answer writes the looped version, keeps it free of NaNs when a whole tile is masked, and tests it the way a grader will.