I don't believe you can devise such a function that will give guaranteed
accurate results for religious purposes.

The Islamic calendar is a strictly lunar calendar, and the time of the New Moon
can be accurately predicted.

However, my understanding is that each Islamic month starts when the lunar
crescent is first seen (by an actual human being) after a new moon.

Although new moons may be calculated quite precisely, the actual visibility of
the crescent is much more difficult to predict. It depends on factors such as
weather, the optical properties of the atmosphere, and the location of the
observer. It is therefore very difficult to give accurate information in
advance about when a new month will start.

--ron