(07-07-2022 10:18 PM)rprosperi Wrote: [ -> ] (07-07-2022 08:03 PM)Thomas Klemm Wrote: [ -> ]Both simulators of the HP-15C return 9.000000000 as a result of the calculator forensic test:

9

SIN COS TAN

TAN^{-1} COS^{-1} SIN^{-1}

So, not simulations of high fidelity. If this is faked, what else is wrong???

Funny how an accurate answer can make one immediately suspicious of a calculating tool...

For fun, I did the calculation in a bunch of calculators I had close to hand.

All of HP's post Voyager calculators agree on their result, and as one one expect the 11C and DM41x agree with the 15C. The TI Voyage 200 is different again.

And then I thought, what the hell, lets input the the calculation on the Casio CG50 - 8.999999998, Casio fx-991ex - 9.000000007, and the TI-nspire CX II - 8.99999998177 (this agrees with the TI Voyage 200).

I know that the TI and Casio hardware calculate trigonometric functions internally to 15 significant digits. As for post-Voyager HP calculators, I had presumed they calculate trigonometric functions internally to 15 significant digits but maybe they only use 13 significant digits.

Interestingly, the TI Voyage 200 and TI nspire CX II actually shows the 15 significant digits as you copy paste the last value via ANS.

If you look at the results statistically, it's the Voyager calculators that are the outliers and if this was indeed a "calculator forensic test", one could suggest they are the least accurate.

As expected, the one post Voyager HP calculator that had a different result was the HP35s - 8.99999986001. This is most probably due to the fact that the HP35s has issues with trigonometric calculations close to zero. As much as it's in the same ball park as the other HP calculators, it's only in agreement up to the 5th decimal place. For some applications, that's a world of difference, and of course for others it's close enough (many engineering applications only require precision of between 4 and 6 decimal places). It really all depends on how the round-off errors stack up in a chain of calculations of this nature.

And seeing as this train of inquiry started off as criticism of the precision capabilities of certain HP-15C emulators, I thought I'd compare the chain of calculations with Mathematica. And here the results agreed with the TI Voyage 200/Nspire all the way until the last two calculations. The ArcCos result varied slightly in the final 4 decimal places and the final result after the ArcSin, resulted back at 9. Which is in fact, the correct answer if all of the calculations are done symbolically - three trigonometric functions, followed by their inverses should resolve back to the original number. It looks a little too convenient that Mathematica still resolves to 9 even though all previous calculations are only accurate to 15 significant digits, but it's bang on the nose.

In conclusion, the older HP hardware is still pretty accurate considering their lower precision, but the fact remains that they begin to deviate from 9 at the 4th decimal place, the post Voyager HP's begin to deviate at the 6th decimal place, and the TI's come out second from best by only deviating from 9 at the 8th decimal place. But it's the Casio's that win in this particular accuracy test, by only beginning to deviate from 9 at the 9th decimal place. Both the Casios calculate all functions to 15 significant digits under the hood, even though they only display 10 significant digits.

There's nothing particularly scientific or robust in these findings but I do find it interesting in terms of some of the 'truths' we hold onto ref Mr Kahan and BCD, which aren't necessarily as true as they once were in 2022.