You would search the blockchain, analyzing all blocks mined on 2011-04-15 and calculating the worth of every transaction and itemizing all these with a worth inside say 10% of the Bitcoin equal of ($800 + $775) / 2 at some change price typical of that day.
This could almost certainly produce a listing of many candidates.
Even when you have been fortunate and there was just one transaction matching your standards, There isn’t any dependable method to uncover something from the transaction knowledge that will assist recuperate management over that cash.
In 2011 you might need positioned the cash into a traditional self-custodial Bitcoin pockets in your PC. In case you not have that PC, or backups of the pockets, there is no such thing as a method to recuperate management over the cash.
Even when you traced it to an change, that change would possibly not exist. it is likely to be tough, or unattainable, to steer any surviving change that you simply have been the official proprietor of cash.
Judging by different questions posted to this web site through the years, the almost certainly final result is that you simply by no means recuperate the $800 however lose tons of or hundreds of {dollars} to faux restoration companies who let you know $800 in 2011 would have purchased perhaps 400 BTC which is now price $23 million.
Postscript
Blocks dated 2011-04-15 (UTC) with an output between 300-900 BTC:
Discovered 31 transactions with an output in vary