hm, why not Chebfun? Result is a rational polynomial so we have to divide, but that seems fine/fast on servers.
[1]: https://www.jstor.org/stable/2157229 [2]: https://www.sollya.org/releases/sollya-8.0/sollya-8.0.pdf
May not have been merged yet.
Pade approximants are also less useful than you might think - it's very hard to get to truly correctly rounded functions with the division.
Fair point about the correct rounding. We'd be fine with several ULPs.