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
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).