Great question. First off, worth mentioning there are many performance metrics you might want to measure. On one hand start up times, but since you specifically mentioned Save, I assume you want to learn more about scroll perf (i.e.: dropped frames, memory usage, time to render next pages, etc).
We don't have numbers we can share but but the React Native team is working on further improving this. Check out Spencer's commit (https://github.com/facebook/react-native/commit/a3457486e39d...) for a sneak peek of what's coming up :).