I think what you want is the supplementary information, part II "completeness proof sketch" on page 12. You already spotted the formulas for "exp" and real natural "L"og; then x - y = eml(L(x), exp(y)) and from there apparently it is all "standard" identities. They list the arithmetic operators then some constants, the square root, and exponentials, then the trig stuff is on the next page.

You can find this link on the right side of the arxiv page:

https://arxiv.org/src/2603.21852v2/anc/SupplementaryInformat...