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: 
The method described in [Hallerbach (2004)](http://papers.ssrn.com/sol3/papers.cfm?abstract_id=567721) always worked well for me.

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> We derive an estimator for Black-Scholes-Merton implied volatility that, when compared to the familiar Corrado & Miller [JBaF, 1996] estimator, has substantially higher approximation accuracy and extends over a wider region of moneyness.
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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)