I'm not even sure I want to know why you have 5/8" gaps around your tub. Home Depot sells a product called backer rod that is a round foam material, sold in a roll, in different diameters. It is designed for that exact purpose, to fill large gaps before caulking. You stuff it in the gap till its lies just below the surface, and caulk over it. It reduces the amount of caulk needed and keeps the caulk from falling through the gap. That's what you need.
Also, it would be wise to fill your tub before you caulk around it, That will make the gap reach it's widest point. If you don't the caulk will set, then next time you fill the tub it will pull the caulking apart or at least stretch it.