Small challenge

[Albert Chan]

(04-23-2023 06:41 PM)John Keith Wrote:  The exact value of 23^10 is 41426511213649, and we can see that a 1 ULP error in the least significant digit ( 4142651121365 instead of 41426511213649) can propagate to give the answer 4.14265112137 rather than the correct answer 4.14265112136.

With halfway round-to-even, it will take 2 ULP error to cause incorrect result.

Quote:However, if 23^10 is computed with 15-digit precision internally, this error should not occur.

x^y = 10^(y*log10(x)), 1 ULP exponent error will cause incorrectly rounded 12-digits result.

log10(23) ≈ 1.3617 27836 01759 28788 ...

23^10 ≈ 10 ^ 13.6172783601759 ≈ 4.14265112136463E13
23^10 ≈ 10 ^ 13.6172783601760 ≈ 4.14265112136558E13

Even if we pull out exponent (to reduce errors), we end up the same error.

23^10 = 2.3^10 * 10^10

2.3^10 ≈ 10 ^ 3.61727836017593 ≈ 4.14265112136491E+3
2.3^10 ≈ 10 ^ 3.61727836017594 ≈ 4.14265112136501E+3