While playing around with the calculator, I stumbled onto the very close identity, where ln(4) plus royal cubit is very close to inverse of the royal cubit.

A bit of rearranging of this identity, replacing the royal cubit by x, allows us to solve for x using the high-school quadratic formula, and thus get a very close approximation for the royal cubit… there are five zeroes after the decimal point if we subtract the royal cubit from the answer.

Here’s the maths: (click to enlarge)

Getting the royal cubit from a simple quadratic equation.

The difference between x above and ₢ (as π/6) comes out at -0.00000727677051.