The introduction calculation was quite long, let’s try something simpler...Using a nice calculation (no carries) results in a perfect answer \[\alpha=1.23456 \to \overline{\alpha}=2.14365 \\ \beta=0.10342 \to \overline{\beta}=1.03024 \\ \alpha+\beta=1.33798 \to \overline{(\alpha+\beta)}=3.17389 \\ \overline{\alpha}+\overline{\beta}=3.17389 \to \overline{(\overline{\alpha}+\overline{\beta})}=1.33798\]
Obviously any error would come from carrying forward and updating the wrong power of the base.