Is there any chance that this might do what you want

mod(239,floor(2 pi,10^-700))

That tries to do mod on 239, over the first 700 digits of 2 pi.

I just randomly picked 239 and chose 700 digits because you mentioned 700 digits above.

Before you trust this I urge you to try this with much smaller numbers and see if you get exactly the result that you are expecting. Please test this multiple different ways to try to make sure this is correct. The limited testing that I have done seems like it is doing this correctly.

The number 5.319372648326541416707296656673541083813*10^255 is a floating point number, with an accuracy of something like 10^216. For such a number it makes no sense to ask mod(x,2Pi), if it is divisible by 6, its cosine, or similar things.

Thank you! Seems to work. What's irritating, however, is that the result of mod( r, floor(2π, 10^(-n))) for r being some 10^260 stays constant from n = 300 to 1100 digits of 2π. Looks like a precision limit in WA to me. Do you have an idea?