Geranium macrorrhizum is sometimes referred to as Big-Root Cranesbill. This mat-forming species does indeed have what can be considered to be a big root/rhizome. Not sure if either description is entirely accurate because roots/rhizomes are subterranean. I separated several specimens from a large patch in another garden to replant in my own. Their long stems were mainly on the surface, more like stolons in that regard.

 Big-Root Cransebill is renowned as a tough plant that will grow in sun, part shade and full shade including dry shade. Most of the common varieties flower in hues of pink but the ones I liberated are G. macrorrhizum 'Album' which I particularly like. The flowers are white with the merest hint of pink. 

 G. macrorrhizum is native to Southeast Europe. There is also a pure white variety sold as 'White-Ness' which is said to originate on Mount Olympus no less.