I am trying to obtain the reciprocal of a floating point value $x$ using the Chebyshev approximation, where $x$ is mostly in the order of $10^3$ to $10^5$. Subsequently, I am trying to implement that algorithm using CKKS for secure computation. My initial attempt with Taylor's series didn't help me that much due to accuracy issues; hence I am looking for alternate solutions. I am trying to obtain an algorithm based on Chebyshev approximation to achieve the same.
I found a plaintext implementation in this link: https://github.com/usnistgov/ChebTools. However, as per the unit testing, the reciprocal test case fails. Simunatnesouly, some computations are not possible to implement using CKKS. I also found an openFHE library here: https://github.com/openfheorg/openfhe-development/tree/main. I believe this library contains an implementation of division using Chebyshev interpolation. However, I could not find the algorithm from this library. It will be really helpful if a division algorithm using Chebyshev approximation can be found. Your help in this regard is very much appreciated.
