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)
Jaeckel has a paper "Let's be rational" in which he *"show how Black’s
volatility can be implied from option prices with as little as two iterations to maximum attainable precision on standard (64 bit floating point) hardware for all possible inputs."*.

I guess it doesn't qualify as closed-form for you, though one might argue that having to apply a deterministic algorithm twice to get accurate answer with machine precision, sort of is.

FWIW, I've not tried to implement what Jaeckal did in "Let's be rational" yet, but I have implemented his previous paper "By implication", which always worked well for me (but relies on a root-search without any guarantees on how quickly it converges).