Should I inform my boss?

About a month ago a coworker brought up an error I had made to my boss. As humilated as I was I decided that I needed to pay more attention to detail as a result of this. I was angry because this coworker and I work together daily and when she makes errors I correct them without making a scene or even telling her... show more About a month ago a coworker brought up an error I had made to my boss. As humilated as I was I decided that I needed to pay more attention to detail as a result of this. I was angry because this coworker and I work together daily and when she makes errors I correct them without making a scene or even telling her sometimes. Of course, I could not use this argument with boss so I decided to start documenting. In the last 30 days this coworker has made 13 errors. Each error I alert the coworker to make corrections and return to me via email. Instead of pointing this out to my boss I alerted my coworker that there were quite a few errors being made on their end. I said this in the most tactful way. Now my coworker is saying that I changed their information to look like they've made these mistakes. I've got solid proof including emails from her that this is untrue. She is threatening to report me. Should I tell my boss first and show her the emails or wait and see what she does first?
Update: Thanks for the great advice. Unfortunately the errors I find I have to document before I correct because if I dont it looks like I was tampering with information. And if I correct it wrong then it all falls on me. I work in accounting and we get audited regularily, She is not the only person I document errors from... show more Thanks for the great advice. Unfortunately the errors I find I have to document before I correct because if I dont it looks like I was tampering with information. And if I correct it wrong then it all falls on me. I work in accounting and we get audited regularily, She is not the only person I document errors from she just happens to have the most errors. Most everyone else appreciates wjhen I ask for correction because they know we are all humans and make mistakes. Plus its inevitable that if its not fixed now it will bite them in the butt later. So I'm definitely not just targeting her.
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