how is Mathematica on computation?
It's great. Especially in domains where it is strong.
If I programmed the math algorithm in Mathematica will it be as efficient as a Xeon processor crunching numbers.
This compares apples and oranges. A compiled, special-purpose algorithm might be beat Mathematica (possibly by a great factor), unless Mathematica already uses a compiled version of the same algorithm. Since WL is an expression rewriting system (sometimes WL is called an "interpreted" language), there is the potential for a significant speed-up in a compiled version of the same algorithm. WL can compile some codes for you. Since WL can run on many processors, comparing it to a processor it can run on doesn't make sense to me. Of course, a bad algorithm on a good processor can be a lot slower than a good algorithm in Mathematica on an ordinary processor.
Sometimes Mathematica's symbolic processing, which is arguably slower than crunching numbers, allows one to take shortcuts that make it much faster than crunching numbers.
I need to know if Mathematica can handle the computational problem of 200+ digits
It's system dependent more than Mathematica-dependent. The theoretical limit on the number of digits of a real number is around 10^15 or more. For me,
on my system (Macbook M4 Max 36GB), I can get 10^10. But adding 1 and 1 with that many digits takes 150+GB, and I can do it only once. Perhaps if I set $HistoryLength to zero, I could do more. But I don't want to work that close to the system limits, so I see no point in trying it out. I can usually get by with 10000 digits or less (normally a lot less), and there's no real obstacle in doing that.
For integers, it's similar, although I think there's no hard limit on how larger integers can be. Whatever the system can take. I can compute 2^(2^40). Once. If I do it twice, the kernel crashes.
