Around 2014 , I proposed a few Bachelier implied volatility “solvers”. The first one was inspired by Steven G. Johnson’s Faddeeva package for the complex error function: it used a piecewise Chebyshev polynomial representation to have a near machine accurate representation of the Bachelier implied volatility. Why did I put “solver” in quotes? Because the problem can be reduced to a 1D function representation (as originally shown by Choi, Kim and Kwak in 2009). Then, I was encouraged by Gary Kennedy to find a simpler rational function approximation, I used 3 or 4 rational functions to cover the full range of the Bachelier implied volatility function.
