Accuracy summation question

[Albert Chan]

(09-07-2023 08:38 AM)rkf Wrote:  It´s indeed strange (altough I would say it differs in the seventh place, ...487 vs. 509).

Assuming your numbers are correct:

log10(3.31627487E5735) vs log10(3.31627509E5735)
=      5735.52065052       vs      5735.52065055

For 12 decimal digits, 1 .. 2000 log10 sum, error is just 3 ULP.

But we are mapping sum with 10^, IP goes to exponent, FP goes to mantissa.
FP with 7 digits accuracy translate to about 7 digits mantissa with 10^, as expected.

10^(FP*(1+ε)) = 10^FP * e^(ln(10)*FP*ε) ≈ 10^FP * (1 + ln(10)*FP*ε)

0.5207*ln(10) ≈ 0.5207/0.4343 ≈ 1.20      // FP → mantissa relative error expanded a little.