The Arabs actually had much more interactions with the Western Europeans than the latter had with the Byzantines, so much more old books have reached Europe through the Arabs. Most Byzantine books have reached Europe only after the Fall of Constantinople and the invention of the movable-type printing press, when their importance has soon become more historical than practical, due to the rapid advances that have started at that time.
Besides the ancient Latin and Greek books, the Arabs have also passed knowledge that had come from India and Persia and also the results of original research that went beyond what had been transmitted from others, especially in chemistry and mathematics, hence words like alchemy and algebra.
However, the works of Avicenna and others like him have reached Europe through Arabs, frequently through Spain.
If you know of another path to Europe for the Persian Arabic works, I am curious to hear which is it.
And “algorithm” too, named after the scientist Al-Khwarizmi
Alabama?
Good guess, but no, it's California from Khalifa.