[VA] SRC #012c - Then and Now: Sum

[Albert Chan]

(12-04-2022 11:58 AM)J-F Garnier Wrote:  What I don't understand in your reasoning is how you moved from:
S*(1-LN2) ≥ sum((G-LN2)/F)
to
S*(1-LN2) = sum((G-LN2)/F) when the sum has converged.

sum index is from 1 to K-1, but we never specify what K is.
If RHS converge, it meant K can be as big as we wanted (literally, K = infinity)

Another way to look at this:

Last dropped term contributed ≤ 1/2 ULP (definition of "converged")
Because of (G-LN2) shrink rate of O(1/2^n), next dropped term ≤ 1/4 ULP, next ≤ 1/8 ULP, ...

Adding effect of growing F, maximum sum of all missing terms will never contribute 1 ULP.
Numerically, we can turn inequality into equality.