Improving precision of VonMisesDistribution CDF?

Questions
1

In comparing the computation of vonmisesDistribution CDF using

N[CDF[VonMisesDistribution[0, 1], -0.78539816339744828], 17]

0.25614866519978797

I can integrate the PDF using

NIntegrate[ PDF[VonMisesDistribution[0,1],x], {x,-\Pi, Divide[-\Pi,4]}]

and by asking for more digits get a value
0.25615930008908783
that agrees with my calculation in C++ using a Boost tanhsinh integration, and with values computed by R and Python with machine precision (17 decimal digits).

I suspect that the integration is terminated after about 6 decimal digit precision rather than my expectation that it will use machine precision of 17 decimal digits.

I have experimented with trying to set $Tolerance, and $MaxExtraPrecision and $MaxPrecision but without success.
Can anyone please confirm this diagnosis and perhaps tell me how to force computation of machine precision 17 decimal digits.

2

Not an answer but there is nothing in the documentation of CDF about modifying the default precision (WorkingPrecision, etc.).

Using NProbability is an alternative to explicitly integrating the pdf (but under-the-hood it might be identical to using NIntegrate):

NProbability[x < -\[Pi]/4, x \[Distributed] VonMisesDistribution[0, 1], WorkingPrecision -> 30]
(* 0.256159300089087906357472644085 *)
3

On Mathematica 13.2.0,I try:

   $MaxExtraPrecision = 1000; N[CDF[VonMisesDistribution[0, 1], 
    Rationalize[-0.78539816339744828, 0]], 100]

   (*0.2561593000890879239407202876346412947984606597806103767750650819525095042102102174133395496879698984*)