Difference between poseidon optimized implementations

[RadNi]

I initially thought the optimized poseidon implemented here follows what it's described in Appendix B of the paper. Which is also outlined here in more details. Iden3's implementation also follows the same description.
But I think the version implemented here is different than the others. I was counting the number of multiplications and additions that we're using in total. According to my calculation, this version has R_p + 1 - t^2 - t less additions and t^2 more multiplications.
Can someone explain what's the optimization that's used here?

[mschof]

The optimized implementation detailed in the original Poseidon paper and in the provided link is only helpful when considering pseudorandom matrices. The multiplications with those would require ~t^2 operations with a naive implementation, but can be made more efficient in the internal rounds with this optimization.

In contrast, Poseidon2 does not use entirely pseudorandom matrices. In particular, the matrices in the external rounds are defined deterministically and the matrices in the internal rounds only have pseudorandom values in the diagonal. This allows using a far more structured approach for the matrix multiplication (see Section 5 in the Poseidon2 paper). With these modifications, the matrix multiplications become more efficient even without the optimization mentioned above (it would actually make the multiplication in the internal rounds slightly more expensive).

Note that this only applies to Poseidon2. The first version of Poseidon technically demands that the same matrix is used in the external and in the internal rounds, and that this matrix is MDS. Neither is true for Poseidon2.