Do you want to update all records in the correlation table? Like this:
UPDATE correlation
SET corr = (SELECT ROUND((COUNT(*) * SUM(x.rating * y.rating) - SUM(x.rating) * SUM(y.rating)) /
(SQRT(COUNT(*) * SUM(SQUARE(x.rating)) - SQUARE(SUM(x.rating))) * SQRT(COUNT(*)
* SUM(SQUARE(y.rating)) - SQUARE(SUM(y.rating)))), 2)
FROM users AS x
INNER JOIN users AS y
ON x.itemID = y.itemID
WHERE (x.userID = user1) AND (y.userID = user2)))
Or
UPDATE c
SET corr = z.corr
FROM correlation c
INNER JOIN (SELECT ROUND((COUNT(*) * SUM(x.rating * y.rating) - SUM(x.rating) * SUM(y.rating)) /
(SQRT(COUNT(*) * SUM(SQUARE(x.rating)) - SQUARE(SUM(x.rating))) * SQRT(COUNT(*)
* SUM(SQUARE(y.rating)) - SQUARE(SUM(y.rating)))), 2) AS corr,
x.userID AS user1, y.userID AS user2
FROM users AS x
INNER JOIN users AS y
ON x.itemID = y.itemID
GROUP BY x.userID, y.userID) AS z
ON z.user1 = c.user1 AND z.user2 = c.user2