The word algorithm comes from al-Khwārizmī. From Wikipedia, I can read that he did a great work on algebra, but I could not find any algorithm attributed to him. Has he actually made any algorithm?
Perhaps his most sophisticated contribution was solving quadratic equations. However, he didn't write the roots of an arbitrary quadratic as functions of its coefficients in the way we do today, because that would require conceding negative numbers exist and can be the arguments of functions. in other words, he classified several types of quadratic, each solved a different way (if roots were conceded). If that sounds odd, remember that centuries later Europe's doubts about negative numbers motivated calling the square roots of negative numbers imaginary, and the different types of quadratic distinguished in the above link were still considered separately. (The eventual motive for taking imaginary/negative numbers seriously was this).