Can a person do the shinigami eye deal without losing half of his lifespan? Let's say, Bob has a death note. He gives up ownership, so that another one with shinigami eyes can tell him his lifespan. Bob is a genius, so he can calculate his death date. He writes his own name in it. 

Bob Hyrule, Bradykardie, 2.3.2034. 

He set his death date, by killing himself with a disease. Now he makes the shinigami eye deal. Will he die earlier?
This will not work due to the 23 day rule.

How to use XXVII

If you write, die of disease for the cause of death, but only write a specific time of death without the actual name of disease, the human will die from an adequate disease. But the Death Note can only operate within 23 days (in the human calendar). This is called the 23 day rule.


Now lets say the person's lifespan will end in 23 days or less. Then the question could be rephrased as What would happen to a person whose name is written in a DN, but whose lifespan is recalculated to be before the date written in that DN? This could be caused by both the shinigami deal or by indirect influence of other people's deaths by DN. In this case we would again end up at the situation where a person's lifespan is before the date set in the death note and as we know from this answer, it is impossible to increase your lifespan by killing yourself with the death note.
How to use VLII


In the Death Note, you cannot set the death date longer than the victim's original life span. Even if the victim's death is set in the Death Note beyond his/her original life span, the victim will die before the set time.


Therefore I'd assume Bob to die within 11 days and a half.