[WIP] Compensated Sum for Global Coordinates

[ax3l]

Long sums, e.g., s of the reference particle, suffer dramatically from floating point precision errors as large and small terms are added in succession thousands to millions of times. Using a compensated sum can keep these numerical errors in check.

This implements, with some vibe coding assistance, the 2nd-order method of Klein (2006) that is linked on Wikipedia. The cost is that s is now of the size of 3 instead of 1 double, e.g., when copying to kernels. We could also use a first order method, which only needs 2 doubles (e.g., Neumaier (1974)).

The latter might be worth considering if we need to use compensated sums for further properties, e.g., x,y,z,t(,px,py,pz,pt) of the reference particle...?

Originally inspired by #1183, but needed at many locations in ImpactX, e.g., existing survey plots #1040, inserting elements #877 #1095 etc. Exposed to Python and used there, too.

To Do

• define a test, tracking errors in s, x, pt, t, etc. of the reference particle -- .e.g, LHC size and beam lifetime
• decide if 1st (Neumann) or 2nd order (Klein) is sensible to use

[ax3l]

Interesting. Should we just use

Suggested change

	iteration.setTime(ref_part.s.value());
	iteration.setTime(ref_part.t.value());

?

(Should be a separate PR, just noticed here...)