Inexpressibility in Exp-Minus-Log (EML)

TL;DR - all numbers defined in EML are computable (mainly because exp and log are computable for computable complex inputs, and the primary input ('1') is ofc computable) - this isn't as obvious as first thought, and you need some machinery from computable analysis. Ultimately, you get the canonical example of Chaitin's \Omega_U is inexpressible in EML (it's left-c.e. but not computable).

EML is also shown to be equivalent to the EL numbers due to Chow (1999). Additionally, the expressions for x*y, -x, and x^{-1} are optimal (there are no shorter EML expressions for those, by exhaustive search).