The only real place for quad precision is when you're working with numbers that truly require you to be able span a massive difference in magnitudes without round-off error accumulating.

For some context, an electron "has structure" roughly on a length scale of about 10^-15 meters (details here unimportant). If you're representing a quantity in O(1) meters, then doing a handful of floating-point operations in double will likely accumulate error on the order of O(1) electrons. Incredible accuracy!

I think the key here is to be mindful of units and make sure you're combining like with like and scaling things appropriately as you go. If you're doing things the wrong way, quad will punish you eventually, too.

An underappreciated tool for working more confidently with floats is interval arithmetic. Worth checking out if you actually (truly) need to do things robustly (most people don't).