This seems to be part of what's going wrong in #33156:

sage: R.<x> = GF(31337)[]
sage: f = R.random_element(degree=999)
sage: g = R.random_element(degree=9999)
sage: S.<y> = R.quotient(f)
sage: %timeit g(y)
188 ms ± 1.74 ms per loop (mean ± std. dev. of 7 runs, 1 loop each)
sage: %timeit (g%f)(y)
16.4 ms ± 31 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)
sage: %timeit S(g%f)
2.97 ms ± 185 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)

This patch adds an optimized code path for evaluation at generators of (quotients of) polynomial rings isomorphic to the polynomial's parent. With the new code, g(y) is about as fast as S(g).

The following cases are handled: {univariate polynomial rings, multivariate polynomial rings, quotients of univariate polynomial rings} over the same base.

We also speed up evaluation at monomials in another polynomial ring in a similar way.

Component: algebra

Author: Lorenz Panny, Marc Mezzarobba, Travis Scrimshaw

Branch/Commit: 9a6e4b2

Reviewer: Marc Mezzarobba, Travis Scrimshaw

Issue created by migration from https://trac.sagemath.org/ticket/33165