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I have a MySQL SELECT query that calculates a distance using Pythagoras in the WHERE clause to restrict results to a certain radius.
I also use the exact same calculation in the ORDER BY clause to sort the results by smallest distance first.
Does MySQL calculate the distance twice (once for the WHERE, and again for the ORDER BY)?
If it does, how can I optimize the query so it is only calculated once (if possible at all)?
No, the calculation will not be performed twice if it is written in exactly the same way. However if your aim is to improve the performance of your application then you might want to look at the bigger picture rather than concentrating on this minor detail which could give you at most a factor of two difference. A more serious problem is that your query prevents efficient usage of indexes and will result in a full scan.
I would recommend that you change your database so that you use the geometry type and create a spatial index on your data. Then you can use MBRWithin to quickly find the points that lie inside the bounding box of your circle. Once you have found those points you can run your more expensive distance test on those points only. This approach will be significantly faster if your table is large and a typical search returns only a small fraction of the rows.
If you can’t change the data model then you can still improve the performance by using a bounding box check first, for example
WHERE x BETWEEN 10 AND 20 AND y BETWEEN 50 AND 60. The bounding box check will be able to use an index, but because R-Tree indexes are only supported on the geometry type you will have to use the standard B-Tree index which is not as efficient for this type of query (but still much better than what you are currently doing).