You didn't comment on post 82 about adding Rd (diode resistance) to make the model complete. (Don't you just hate variables?)
So using your math model add .53 ohms to the large diode in my simulation, in post 82, and 1.8 ohms to the small diode and see if the curves don't come very close. Without Rd the curves go almost straight up (as you have pointed out) after a small current flows. If you like that, add 17.5 ohms to the little LED and see if it doesn't match. Pretty cool.