Hello.
Have a look at this if-block

if(1-tEtot/theTarget.GetMass()>0.001)
{
   theTarget.SetTotalEnergy(tEtot);
}

The if-statement could be put as Etot<0.999*theTarget.GetMass() and when theTarget.SetTotalEnergy(tEtot) is called

inline void SetTotalEnergy( const G4double en )
{
   totalEnergy = en;
   kineticEnergy = totalEnergy - mass;
}

it will set totalEnergy less than mass and negative kineticEnergy which could possibly affect low energy neutron transport at low temperatures.

I have not checked if this behaviour really appears during the simulation but it would be better this way:

if(tEtot<theTarget.GetMass())
{
   theTarget.SetKineticEnergy(0);
}

The bug is certainly there, but actually does not cause a round-off error.  The conditional in the if statement is incorrect:  it should be tEtot/theTarget.GetMass() - 1 > 0.001  .
This way the relativistic energy is chosen at higher energies while the classical energy is chosen at lower energies. I believe the classical energy is used to maintain precision at low energies, as the sqrt would reduce the number of significant digits.
A fix has been submitted.