I am trying to find achievements of Islamic society in the realm of science. However, all I coming up with is mostly that they translated Greek classical work into Farsi/Arabic. One related claim is that without Muslims the Greek work might have been lost.
Is there a definitive list of substantial scientific achievements during Golden age of Muslim rule?
As a preface, I think it is important to understand that "mostly translating" ancient texts was in fact quite important - Scott Montgomery's Science in Translation is a good, albeit slightly dated, introduction the history of scientific translation, and does discuss Golden Era translation for Greek astronomy.
As for more direct scientific accomplishments, Avicenna's The Book of Healing and The Canon of Medicine were essential medical texts until the 16th century. Aristotle and Galen are central to these works, but in addition to preserving them Avicenna reinterprets and updates them with contemporary knowledge. al-Razi similarly made direct contributions to Galenic medicine. Averroes and Avenzoar wrote extensively as well on medicine, and al-Khawrizmi is quite famous for his work on equations that would become the basis of algebra.
Much later, Ibn Khaldun's Muqadimmah is a very rich, expansive text covering everything from historiography to evolution. This is a very loose sketch of Islamic contributions to science, but Golden Era scholars did both preserve Classical authors and make influential contributions themselves.
Episodes in the Mathematics of Medieval Islam (by J. L. Berggren) is a good introductory book specifically for mathematics. It was a prescribed reading from an undergraduate course from the Islamic department of the university. The book goes over the different fields and theories of mathematics that were developed as well as the names of the mathematicians who developed them. The fields range from arithmetic to number theory and combinatorics.
An interesting fact about Islamic science is that it also actually contains the science of the Quran and Hadith and other forms of religious science. I am assuming you mean natural science in your question, so I won’t go into much details on religious sciences. If you are interested to learn more on the classification, I would advise Al-Ghazali’s The Revival of the Religious Sciences.