Many general purpose processors (including Intel's) may not always produce the correctly rounded result of a floating-point operation due to double rounding. Instead of rounding the value to the working precision, the value is first rounded in an intermediate extended precision and then rounded in the working precision; this often means a loss of accuracy. We suggest the use of rounding to odd as the first rounding in order to regain this accuracy: we prove that the double rounding then gives the correct rounding to the nearest value. To increase the trust on this result, as this rounding is unusual and this property is surprising, we formally proved this property using the Coq automatic proof checker.