Free42 possible accuracy flaw

[Albert Chan]

(03-30-2022 08:04 AM)Werner Wrote:  531441^1072.5 vs 9^6435

The former is off by 2 ulps.

Normally, we would not expect 2 ulp error or more.
(except when exponent dropped, say from 10 to 9.999...)

Measure in fractions of ulp, from exact result, we expected EXP part 1/2 ulp or less,
correction part of 1/2 ulp or less. Combined error almost always less than 1 full ulp.

For above example, log of result gives:

14139. 14015 51585 71728 25680 61991 92905 81881 ...

Here, log result had huge integer part, which kills the precision.
If we had 128 good bits, (about 38 digits), say 81881 → 81900, we recovered 33 digits:

81881 → 3.5526 15792 57691 44387 92654 15646 88743 56604E+6140
81900 → 3.5526 15792 57691 44387 92654 15646 88750 31601E+6140

Even with correct recovery, EXP *then* ROUND, we were off 1 ULP.
The flawed recover part added another possible ± 1 ULP