Double rounding consists in a first rounding in an intermediate extended precision and then a second rounding in the working precision. The natural question is then of the precision and correctness of the final result. Unfortunately, the used double rounding algorithms do not obtain a correct rounding of the initial value. We prove an efficient algorithm for the double rounding to give the correct rounding to the nearest value assuming the first rounding is to odd. As this rounding is unusual and this property is surprising, we formally proved this property using the Coq automatic proof checker.