Your formula doesn't work. There have been 7 leap days including 1984. The date of the leap day is relevant to your formula because if he were born on Jan 1 there would be a difference in your result. Also, May 14 plus 138 days would have been September 29, not Sept 30.
In the this case 'to' means 'through' and is inclusive of all the days. By adding the difference we do not twice count the number we arrived at from the first part of your calculation.
