I am developer of Advanpix toolbox. You mentioned it in the comments and that is why I decided to respond.
There are several misconceptions regarding computation of special functions (especially difficult ones, like Bessel or Hypergeometric or alike).
(a) First one is that such functions are considered easy to compute. You can frequently see the suggestions like - just use series, Pade, Chebyshebv or some other approximations.
(b) Second one is that well established software (NAG, MATLAB, Python, Julia, etc.) provide high-quality implementation for such functions.
Both are false. We have been working in the area for quite some time and published some of our findings, e.g. see [1,2]. Even double-precision software fails miseably to deliver proper accuracy.
When it comes to extended precision the situation gets even more difficult. No fixed degree approximation can be used, accuracy must be tracked on the-fly during the computations, cancellation effects must be detected, overflow/underflow must be handled properly, etc. etc. Combine this with different kinds of arguments and orders (complex, pure imaginary), principal values and branch cuts - and this can easily become the subject of solid research for several years.
Luckily you don't have to do all this, as we already did it for our toolbox. There is a reason why profesionally developed software costs money - it simply saves you ton of time by providing ready-to-use high-quality functionality with guaranteed accuracy.
Another advantage of our toolbox over "free" software is performance. We are several orders of magnitude faster than VPA, HPF, MP or even Maple, Mathematica, Julia and Python . Multi-theading are natively built-in and used in all operations.
So, if you consider the quality and performance, then it becomes clear that there is no such thing as "free" software. Seemingly "free" software costs you a lot of time due to longer execution times, pain in setting up, etc. But most importanly your research might fail just because some "free" library provided sub-optimal accuracy in computations. That is pretty high cost.
Note 1. We didn't run speed comparison with HPF. But results are clear because of the author claim: "As I showed, my own HPF is at leasrt similar in speed to the use of VPA. There will possibly be some cases where one may be faster, some slower. At least they are of the same order of magnitude, of that I am happy." The VPA is the slowest extended precision software out there, we made comprehensive comparison with it .
Note 2. Using your example, this is how Advanpix toolbox computes it with 100 digits of accuracy on my computer:
Elapsed time is 0.000923 seconds.