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1 ENTER 3 / 3 * 1 -
01-23-2016, 10:00 AM
Post: #21
RE: 1 ENTER 3 / 3 * 1 -
Interesting read, Gerson, thanks for excavating. Smile

BTW, anyone having an idea what happened to Karl S.? Always enjoyed his posts.

d:-?
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01-23-2016, 01:02 PM
Post: #22
RE: 1 ENTER 3 / 3 * 1 -
(01-23-2016 10:00 AM)walter b Wrote:  Interesting read, Gerson, thanks for excavating. Smile

BTW, anyone having an idea what happened to Karl S.? Always enjoyed his posts.

d:-?

Yes Gerson, thanks a lot. And thanks to all who convinced me that it's not just an HP phenomenon.
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01-24-2016, 02:50 AM
Post: #23
RE: 1 ENTER 3 / 3 * 1 -
SR-51A produces 1. E-12

--Bob Prosperi
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01-24-2016, 11:11 PM
Post: #24
RE: 1 ENTER 3 / 3 * 1 -
(01-24-2016 02:50 AM)rprosperi Wrote:  SR-51A produces 1. E-12

Did it actually return a positive number, or is that a typo?
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01-25-2016, 02:32 AM
Post: #25
RE: 1 ENTER 3 / 3 * 1 -
(01-24-2016 11:11 PM)hank Wrote:  
(01-24-2016 02:50 AM)rprosperi Wrote:  SR-51A produces 1. E-12

Did it actually return a positive number, or is that a typo?

Typo, it's negative as you would expect. Careless mistake, sorry, I was in a rush to post it as I was happy to get the TI charged and working. While I'm not sure it's right to call it excited, it was nice to have a reason to take out my last pre-HP machine for a small exercise.

--Bob Prosperi
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01-25-2016, 09:50 PM (This post was last modified: 01-25-2016 09:50 PM by Dave Hicks.)
Post: #26
RE: 1 ENTER 3 / 3 * 1 -
(01-23-2016 10:00 AM)walter b Wrote:  BTW, anyone having an idea what happened to Karl S.? Always enjoyed his posts.

Years ago, he sent me several emails asking me to do something about the behavior of one of the forum's top posters. I didn't get a chance to look into his complaints until much later, by which time he had stopped posting. Now we have additional moderators so hopefully we can keep up with these issues.
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01-26-2016, 12:08 AM
Post: #27
RE: 1 ENTER 3 / 3 * 1 -
HP-41C: -1E-10
HP-16C: -1E-10 -- in float mode
Prime: -1E-12 -- in several modes (but not CAS)

Incidently, my 41C started to whine when I tried to turn it off. Momentary battery disconnect cured it, but I hope it isn't a sign of deterioration. It hadn't been powered up for a few weeks. Date/Time unaffected but MEM LOST afterwards.
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01-26-2016, 08:37 PM
Post: #28
RE: 1 ENTER 3 / 3 * 1 -
My favourite is 3^3-27, it's works well on those models which have integer exponentiation routine and do not use the a^b = EXP(b×LN(a)) method.
Also interesting the (SQRT(2))^2-2 calculation.

And my favourite for SOLVE: 3×X+1÷(X-5) = 15+1÷(X-5) and solve it for X. (Hint: X=5 is wrong answer...)
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01-27-2016, 07:21 PM
Post: #29
RE: 1 ENTER 3 / 3 * 1 -
Hello all. Please help me understand this. I want to express my thoughts with diplomacy. With all this discussion of accuracy and precision, most of the discussion seems to gravitate towards inaccuracy and rounding flaws. So, in light of these shortfalls, in what context, in what degree of reliability is a calculator, be it Sharp, Casio, TI or even (forgive me, please) HP?

Like I said, I am trying to make sense of a calculator's practicality and reliability in light of this discussion's findings.

Thanks in advance.
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01-28-2016, 12:17 AM (This post was last modified: 01-28-2016 12:33 AM by Dieter.)
Post: #30
RE: 1 ENTER 3 / 3 * 1 -
(01-26-2016 08:37 PM)Csaba Tizedes Wrote:  My favourite is 3^3-27, it's works well on those models which have integer exponentiation routine and do not use the a^b = EXP(b×LN(a)) method.

All HPs since the HP92 (IIRC) internally use additional guard digits so that 3^3 will return 27 (exactly) and 3^3–27 yields a plain zero. No integer exponentiation routine required.

The three guard digits are sufficient for most cases, yet not for all. Very large results > 10^20 may be off in the last digit. The 15C Advanced Functions Handbook states a possible error within 3 ULP. On calculators with a larger working range (x<10^500) the error may get somewhat larger. On the other hand, integer results within ±999999999[99] should be exact. Including 3^3. ;-)

(01-26-2016 08:37 PM)Csaba Tizedes Wrote:  Also interesting the (SQRT(2))^2-2 calculation.

Sqrt(2) is irrational, so there is no 10-digit (or 12-digit, or 16-digit...) value that exactly equals sqrt(2). The square of this result may happen to round to 2 or not.

On a 10-digit calculator sqrt(2) is returned as 1,414213562. This is the exact value for the true square root 1,4142435623730950488.. rounded to 10 digits.
1,414213562^2 again is 1,99999989447... which in turn is correctly displayed as 1,999999999. A calculator that returns a plain 2 for 1,414213562^2 is simply ...wrong. Simply because there is no 10-digit value which, when squared, rounds to 2 again. 1,414213562^2 yields 1,999999999 and 1,4142135623^2 returns 2,000000002.

Let us not forget: our calculators do not return sqrt(2), or sin(40°), or ln(3). They return a number that resembles the true result as closely as possible (within 10 or 12 digits). But this number is not identical with the true result.

(01-26-2016 08:37 PM)Csaba Tizedes Wrote:  And my favourite for SOLVE: 3×X+1÷(X-5) = 15+1÷(X-5) and solve it for X. (Hint: X=5 is wrong answer...)

My 35s returns 4,99999999999. ;-)

Which, in a way, is the "best" result you can get. For x=5+ε, the difference between the left and right hand side is 3 ε. Since the function is not defined for x=5, the value with the smallest possible ε comes as close to zero as possible.

Dieter
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01-28-2016, 09:54 PM (This post was last modified: 01-28-2016 09:55 PM by Matt Agajanian.)
Post: #31
RE: 1 ENTER 3 / 3 * 1 -
Hinall.

Yes, the foresics are quite useful. But, I am still unclear. Given all these accuracy and reliability isses, in what ways, in what contexts are a calculator and its results practical, accurate, useful, reliable, and trustworthy?
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01-29-2016, 06:52 AM
Post: #32
RE: 1 ENTER 3 / 3 * 1 -
(01-28-2016 09:54 PM)Matt Agajanian Wrote:  Given all these accuracy and reliability isses, in what ways, in what contexts are a calculator and its results practical, accurate, useful, reliable, and trustworthy?

I do not think there are any "issues" here. What a decent calculator returns is accurate and reliable. It's up to the user not to draw the wrong conclusions:

[2] [sqrt] [pi] [x] => 4,442882938

Now listen closely to your calculator and hear what it says:
"I don't know what Pi times the square root of 2 is, but I can tell you the first ten digits of 1,414213562 times 3,141592654 are 4,442882938".

Can you hear it ?-)

Dieter
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01-29-2016, 09:09 AM
Post: #33
RE: 1 ENTER 3 / 3 * 1 -
(01-29-2016 06:52 AM)Dieter Wrote:  I do not think there are any "issues" here.

I think there are some issues. Calculator vendors rarely make any reference to their product as an unreliable source of information (I recall some HPs have notes in their manuals, but even those are relatively rare and infrequent).

If I am analyzing a series of data I would normally form an associated measure of the data error margin -- the classic "ANSWER IS x +/- y" due to uncertainties in the measurements.

When using sliderules, log tables etc, we have an awareness that the calculation process itself is imprecise. Using a calculator DOES, I believe, give an illusion of spurious accuracy. It is certainly common enough to be debated here on numerous occasions.

It would not be beyond wit or wisdom for a modern calculator to perform a similar analysis of error margin (due to calculation) and present the answer in similar form.

My concern would be that in processing a complex calculation, especially in a program, one quickly loses a feel for the error margins implicit in the calculation. One has little idea, without further exploration, of where those digits on the display lose their validity.

A modern calculator could realistically keep track of and display an estimate of the error inherent in the calculation. The algorithms used for various functions and constants have definite limits which could be automatically tallied by the machine.

It is, of course, just an opinion.
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01-29-2016, 10:08 AM
Post: #34
RE: 1 ENTER 3 / 3 * 1 -
(01-29-2016 09:09 AM)ColinJDenman Wrote:  It would not be beyond wit or wisdom for a modern calculator to perform a similar analysis of error margin (due to calculation) and present the answer in similar form.

It is beyond large computers to do this automatically and it is what keeps numeric analysts employed. The reason is that any error bounds explode very quickly and give a useless range as a result. For example: 10±1 ⨉ 12±1 = 121±22. A small error in the last digits will be compounded quickly, more so for functions with steep gradients.

It is possible for algorithms to be designed so that they produce a narrower result but I'm not aware of anything automated that can do this reliably. An additional confounding factor is that, by design, a calculator is always working one step at a time and cannot have an holistic view of the computation.

A good book on the subject is Validated Numerics by Tucker.


- Pauli
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01-29-2016, 04:09 PM
Post: #35
RE: 1 ENTER 3 / 3 * 1 -
Thank you for those three elaborate and detailed posts. They are the perspectives and analyses I was looking for. They concisely and thoroughly explained the necessary perspective to view handheld calculator calculations. I appreciate that.
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02-07-2016, 09:58 PM
Post: #36
RE: 1 ENTER 3 / 3 * 1 -
(01-29-2016 10:08 AM)Paul Dale Wrote:  A good book on the subject is Validated Numerics by Tucker.
- Pauli
Thanks for this reference book :O)
I just found a presentation by the same author that might be interesting to browse, before buying the book.

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