Sage is an open-source computer algebra system with extensive support for computing with elliptic curves. Sage provides both a command-line interface and a browser-based GUI (the Sage notebook). Tutorials and many examples can be found on-line (google is your friend). You can download a copy of Sage to run on your own machine if you wish, or create an account for free on the SageMathCloud™.
Explicit formulas implementing the group operation on elliptic curves in a wide variety of forms can be found here. These formulas have been carefully optimized and verified for correctness.