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The old index x[i1 + d1 - s_in] only works when the matrix has 2 * p_in + 1 diagonals. Diagonal d1 of row i1 is the column i1 - q + d1, where q is the pad of the matrix. That column sits at x[i1 - q + d1 - s_in + p_in], because x is padded with p_in. When q = p_in this reduces to the old i1 + d1 - s_in. When the matrix has fewer diagonals (q < p_in, e.g. a block between spaces of different degree), the old index is shifted by p_in - q, which gives wrong results. That's why those matrices used to raise NotImplementedError.
So off1 = p_in - q - s_in is that shift. It is computed once outside the loop, and for q = p_in it is exactly -s_in, the old offset.
I agree off1 hides what it means. I can write it out instead, e.g.
q1= (mat.shape[1] -1) //2# pads of the matrix
...
val+=mat[p_out+i1_loc, d1] *x[i1-q1+d1-s_in+p_in]
so the column i1 - q1 + d1 can be read directly from the index. Would that be clearer?
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