Cry Me a River, written by Arthur Hamilton, was first published in 1953 and made famous with the version by Julie London, as here: https://www.youtube.com/watch?v=DXg6UB9Qk0o
There have been other vintage versions Fitzgerald, Streisand and Bassey, and more modern ones by Justin Timberlake, Joe Cocker and Michael Bublé.
Play it smoothly, the London way, or sacrilegiously Jazz it up, whatever you fancy!
Chris has kindly supplied dots and backing track (see link below) and, who knows, if I can get my DIY recording set up sussed I may even embarrass myself with a first recording!
The backing track and sheets can be found here "Cry me a River"