$z= log( exp($x) + exp($y) )
      = log( exp($x)*( 1 + exp($y)/exp($x) ) )
      = log( exp($x) * ( 1 + exp($y-$x) ) )
      = log(exp($x)) + log( 1 + exp($y-$x) )
      = $x + log( 1 + exp($y-$x) )