Trigonometric Functions: Argument reduction

Question

I'm trying to implement trigonometric functions for interval arithmetic, preferably with < ulp error which requires precisely compute trigonometric functions manually. The first step is usually argument reduction, but I'm not really sure how argument reduction is usually implemented. For example, with Python, I get the following results:

>>> import math
>>> math.sin(1e100)
-0.3806377310050287

What I expected it to do is apply modulo 2 * math.pi, but this gives me a different result:

>>> math.sin(1e100 % (2 * math.pi))
0.6806339877607344
>>> math.sin(math.remainder(1e100, 2 * math.pi))
0.6806339877607344

How is argument reduction done for trigonometric functions?

Answer 1

It seems to be the case that most implementations simply use high precision with pi. There exists other strategies, but they all boil down to using lots of precision somehow. Once you have a highly precise value of pi, you can simply take x % (2 * pi) and work off of that.