Unknown Plant

I have a plant,with very tiny white flowers, which could possibly be a weed,that has taken up residence under one of my rose bushes. I comes back on it's own every year,so it must be a perennial..I don't mind it being there since it is a nice ground cover,but I would sure like more information on it.

This is a segment from the plant. It is multi-jointed and seems to grow forever from a main root. I'm not sure if it takes root along the branches. I haven't noticed it doing so.It crawls all under my rose bush and has even overflowed onto the steps and sidewalk. It is in a location that we do not use often, so I haven't bothered to trim it back. I did take a few cuttings to see if it would propagate in a glass of water,but no luck yet.I have unintentionally cut a few ends off with my mower when I mow in that area.

I hate to spread it around if it is going to be a nuisance like the Creeping Charlie plant which I thought was so cute. I brought a sprig of it to my home,years ago, and I have regretted it ever since. It has taken over much of my yard, and is unstoppable.I wouldn't mind putting this plant in my big flower garden. That bed is enormous and I could use a ground cover,to keep the undergrowth in check.It is an area which contains huge slab rock that protrude above the ground, so I have encircled the area,mulched it with everything imaginable,added a little compost here and there and have planted intermittent flowers throughout the bed. Some areas,over  the rocks, now have a thin layer of mulch and soil , so I think it would be suitable for a ground cover with shallow roots.I am hoping someone might  have an idea as to the name or growing habits of this plant.