Ubuntu 6.06 To Ubuntu 16.04 LTS Performance Benchmarks: 10 Years Of Linux Performance
With the graphics tests from the Radeon HD 5830, they were only usable going back to Ubuntu 12.04 with the R600 Gallium3D driver. I could have popped in an older graphics card, but still going back that many years in the pre-Gallium3D days the open-source driver performance was incredibly slow that it would have hardly been worthwhile.
Between Ubuntu 12.04 and 14.04 was the biggest boost in performance.
Tesseract only went back to Ubuntu 14.04 due to needing SDL2.
With the R600 Gallium3D driver in a mature state, there isn't much more to see performance-wise out of this graphics card on Ubuntu 16.04. However, there is OpenGL 4.1 and other improvements. Most of the new feature and performance these days for the open-source AMD Linux code is focused on RadeonSI for the GCN GPUs.