HHC 2015 - Savage benchmark curiosity

[Dieter]

(10-08-2015 08:31 PM)Claudio L. Wrote:  Question still remains: why does it self-correct?

I do not think it "self-corrects", it just happends to round once up, once down. Depending on where you stop the loop the results sometimes match, sometimes they don't.

Take a look at tan(arctan/1470.99991143)). The exact arctan is 1,570 1165 1720 5390... Both the 35s and the 39gII, two 12-digit calculators, correctly round this to 1,57011651721. But the former returns the tangent of this (1470,9999 2140 4527...) as 1470,99992141 while the latter displays 1470,99992140. So there is always a chance that the last digit is off by one. I remember the accuracy discussion in the HP15C Advanced Functions Handbook. Here the trig functions are said to return results with an error between 0,5 and 1 ULP. Which means the last rounded digit can be one digit high or low.

If you google "savage benchmark" you will come across a PDF on technicalc.org that shows results for several calculators. It also includes a graph that shows the error after 1...5000 iterations. The essential point here is that the error graph does not continuously drop, as the accuracy of the result after n loops is expected to decrease, but – while generally decreasing – it bounces up and down. So at some points the accuracy actually increases (which is what you may call "error compensation") and at other points it decreases again. Which is what can be expected due to the more or less random errors in the last place that may occur any time. The final error after n loops depends on where you stop – it may be smaller or larger, and even smaller for a larger number of loops. ;-)

Dieter