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: 
There are some other references:

* [Li and Lee (2009)](http://dx.doi.org/10.1080/14697680902849361)
[[download]](https://ssrn.com/abstract=1027282)
An adaptive successive over-relaxation method for computing the Black–Scholes implied volatility
* [Stefanica and Radoicic (2017)](https://ssrn.com/abstract=2908494) An Explicit Implied Volatility Formula

Related discussions on the implied volatility inversion:

* [How can the implied volatility be calculated?](https://quant.stackexchange.com/questions/7761/how-to-compute-implied-volatility-calculation)
* [What is an efficient method to find implied volatility?](https://quant.stackexchange.com/questions/15198/what-is-an-efficient-method-to-find-implied-volatility)

For the normal (or Bachelier) implied volatility, there's an improvement to Choi et al (2009) [[paper](http://dx.doi.org/10.1080/13504860802583436) / [SSRN](https://ssrn.com/abstract=990747)] mentioned in the question:

* [Fabien Le Floc'h (2016)](https://ssrn.com/abstract=2420757)
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.