Torture tests: what do they mean?

[Manolo Sobrino]

(05-14-2014 07:23 PM)Dieter Wrote:  Most HP calculators since the HP-91 work with three additional digits. For internal calculations, that is. These digits are not presented to the user. The input has always just the number of digits you can enter. That's the crucial point here: you would need a 20-digit input for 355/226 in order to get the output (the tangent) correct to 12 digits. As already stated, you cannot trust the last eight digits.

I don't know. Testing the 50G while trying to understand the behaviour of the Prime I found that arguments are passed to functions rounded to 12. That's what they do here, they round 355/226 to 12. If they had considered 15 digits they wouldn't have got those results. There's just no way to reproduce that while rounding 355/226 to 15... In fact, the last 8 digits are accurate for the argument rounded to 12 digits, so you can really trust them if you're looking for tan(1.57079646018). But weren't we interested in tan(355/226)? In such a case the last 8 digits are not accurate. The question is why do you round when you can work with more digits? And if you have to round, what's the point of producing an accurate output for the rounded input which is of no use?

The whole precision issue for the trigonometrics was hard to understand until I found a document from TI explaining what they did for the TI-68. Then it was clear, they were doing the same. Storing Pi to 24 digits.

You can't get more precision from just an algorithm in this case. You need additional information to calculate such a number of significant digits. It has to come from high precision data in the CORDIC tables, 15 digits just don't cut it.

I got the HP results for 19 significant digits as well, probably an effect of rounding the result to 12. But the real number is not 19 or 20, for Sin it's 12*2=24, so it's likely that for all the trigs.

I haven't tested it thoroughly in order to find where they use the guard digits. I like calculators but I have the growing feeling that there's no point in wasting much more productive time with black boxes and marketing sales pitch. I've just found too many bizarre practices that are even more bizarrely rationalised. I would never work with double precision for some things, single precision for others, and then mix them. For me, it's just a lousy way to deal with numbers that has to do with the limitations of the platform and an ill-conceived idea of accuracy.