HHC 2015 - Savage benchmark curiosity

[Dieter]

(10-08-2015 12:34 PM)Claudio L. Wrote:  I think you are right, which makes it even more incredible that some random error in the last digit could "self-correct" back to the original value. I refuse to believe it's a coincidence, there has to be a mathematical explanation for that... but it's beyond my knowledge.

Maybe we can solve this... if (!) I only understood what makes you wonder here.

From your initial post:

(10-07-2015 03:18 PM)Claudio L. Wrote:  The surprise was... it diverged to the exact same value as before!
Result was 2499.99948647, even when starting from the "correct" sequence after value TAN(ATAN(1469.99992223))+1 = 1470.99991143 was used as starting point.

Why is this surprising? because the difference after doing ATAN and TAN on 1469.99992223 was 2160e-8, and for example the next error at 1647.99989715 is 2716e-8. I don't see how the sequence could "self-correct" to give the exact same result? Perhaps somebody has an explanation?

The difference after an tan(arctan x) for x=1,57... here is 1 ULP of x = 1E–11 divided by the first arctan derivative, i.e. 1/(x²+1). Which yields ~ 1E–11*(1470²+1) = 2,161E–5 resp. 1E–11*(1648²+1) = 2,716E–5. Which is what you get. This also shows that a very, very slight error in x (merely one single ULP due to an even lesser error in the 15-digit result) will cause large errors in the next step. And these errors are very similar for x=1469, 1470 or 1471. So if one result is 1 ULP high and one of the next is 1 ULP low this yields very similar errors that may compensate each other.

But maybe I did not understand your surprise correctly. In this case please provide a step-by-step analysis.

Dieter