Question: While the solution for IV can certainly be reached using numerical search methods, I wonder if a high precision closed-form approximation exists.

For example, there is a very robust (precise within 10^-12) approximation for Bachelier IV ([paper](http://dx.doi.org/10.1080/13504860802583436) / [SSRN](http://ssrn.com/abstract=990747)), but is there anything similar for Black'76 and/or Black-Scholes?

Answer: 
Let's Be Rational uses exactly two iterations to give full machine accuracy for all inputs. It can be viewed as a three-stage analytical formula if you like.

The code is free to download at www.jaeckel.org.

Rgds,
Peter
Peter Jaeckel methods from the papers mentioned are the industry standard used by most practitioners to get IV. 

In addition in practice the article you mention is probably of very little use because the analytic approximation you refer to in the SSRN paper needs both call and put price to extract the implied vol however usually only one of the 2 instruments is liquid when you are not close to ATM.