Different trig algorithms in CAS and Home?

[Dieter]

(01-04-2018 02:39 PM)chromos Wrote:  Thank you for reply, Dieter, even it didn't answer my question why current calculator doesn't have better precision (or accuracy - my knowledge of english isn't capable differentiate these terms) than calculator 40 years old.

I think there is an answer. ;-) The question is whether the TIs have an advantage or not. This requires some insights on how TI and HP calculators work.

Since about 1976 HP's calculators internally work with (usually) three so-called guard digits, i.e. they carry out their calculations with 3 more digits than returned to the user. So the HP67 or HP41 display 10 digits, but they use 13 internally. This is required to ensure full 10-digit accuracy for the final result, and in most cases this goal is met. So once the calculation is done, the result is rounded to 10 digits, and this rounded result is displayed.

The TIs follow a different approach. The TI58/59 also use 13 digits for their calculations, just as the HPs do. But they do not round their results before they are returned to the user: instead they give you all 13 digits. Which means that the last one or two may be off.

Example: try 3^4 on the TI58/59. The 10-digit display will show "81" - fine. But the result actually is 80,99999999996. The HPs probably have not got a better result with their internal 13-digit calculation, but they round it off to 81 (10 digits) before it is shown to the user. These are simply two different philosophies: both TI and HP used 13 digits for their computations. While TI chose to return all 13 - knowing that the last one or two may be off - the HP approach was to play safe and return only what (most probably) was exact.

Both philosophies have their benefits. I like the idea of having 13 digits not only internally, but for all user calculations. Even if the result is not dead on (it can be rounded to display precision with a EE INV EE sequence). On the other hand I also appreciate the HP way of returning only reliable 10 digits.

This also explains your observation that 1/9 - 0,111111111111 yields zero. Of course it does: the returned 12-digit value of 1/9 is 0,111111111111, so the difference is indeed zero. Remember that the 15 digits are only used internally and not exposed to the user.

(01-04-2018 02:39 PM)chromos Wrote:  How do you know the HP Prime works with 15 digits internally?

Well, all HP calculators with 12-digit BCD arithmetics I know of use 15 digits internally, just as the 10-digit devices use 13. So I think it's safe to assume that the Prime doesn't behave differently here. ;-)

But you can do a test: Try sin(3,14159265359) in radians mode. The TI58/59 returns a plain zero here. The common 12-digit HPs give the correct result -2,06761537357 E-13. Which shows that pi internally is stored with much higher precision than 12, 15 or even 20 digits.

Dieter