The previous code used an equality test for each index. This variant uses a "MUX tree" instead. If we imagine the items as being the leaves of a binary tree, we can compute the `i`th item by splitting `i` into bits, then performing a "select" operation for each node. The bit used in each select is based on the height of the associated node.
This uses fewer wires and is cheaper to evaluate, saving 31 wires in the recursion circuit.
A potential disadvantage is that this uses higher-degree constraints (degree 4 with our params), but I don't think this is much of a concern for us since we use a degree-9 constraint system.
Just passing the "combined constraints" buffer into `eval_filtered_recursively`, so that we can combine a mul by the filter with an add into the buffer. Saves 56 wires.
* Specialize `InterpolationGate`
To cosets of subgroups of roots of unity. This way
- `InterpolationGate` needs fewer routed wires, bringing our minimum routed wires down from 28 to 25.
- The recursive `compute_evaluation` avoids some multiplications, saving 100~200 gates depending on `num_routed_wires`.
* Update test
* feedback
PoseidonGate's recursive evaluations were using a lot of gates, and the MDS layer was the main culprit.
The other issue is that `constant_layer_recursive` creates a bunch of `ArithmeticGate`s with unique constants. We could either change `ArithmeticGate` to support different constants per operation, or wire in constants from `ConstantGate`, and change `ConstantGate` to support several constants per gate.
This won't really help anything near term since we're still between 2^12 and 2^13, but could have some benefits later, depending on what recursion arities and security settings we end up using.
`PoseidonMdsGate` needs `2 * D * WIDTH = 48` routed wires, and the combination of adding a gate and increasing routed wires slows down the prover a bit. So for now, I kept it at 28 wires, and the old code path is still used.
* Batched eval_vanishing_poly_base
* Reduce the number of allocations
* Lints
* Delete unused things
* Minor: fix a debug_assert
* Daniel PR comments
* Lints
* Daniel PR comments
