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Let's denote the Implied normal volatility (Bachelier) as $\bar{v}$, and the implied log-normal (Black Scholes) as $v$.

When everything else is known (spot, strike, maturity, rates etc) how can you go from $\bar{v}$ to $v$ and vice versa?

The math suggests that there must be a simple formula you can just plug but unfortunately I cannot find that formula. Of course I know that I can derive it myself, but I am just interested in the result. So the answer does not have to include the proof derivation.

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