Pioneer vs RPL precision

06072022, 02:26 PM
Post: #1




Pioneer vs RPL precision
Yesterday I was playing with EMU42 and realized that is some circumstances Pioneer calculators (no matter if RPN or algebraic) shows better precision than RPL models (from 28s to 50g).
The operation is the classic "(10/3)*3". On any Pioneer and even 35s the result is always 10, even if calculated as a program or an algebraic expression On any RPL model the result is:  9.9999999999 if calculated as "10 <ENTER> 3 / 3 *"  9.9999999999 if calculated as << 10 3 / 3 *>>  9.9999999999 if calculated as '10/3*3" <EVAL>  10 if the operation order is changed, i.e. '3*10/3', regardless the way it is calculated The reason of RPL models behaviour appears logic to me, as the result of 10/3 is approximated and the error is propagated to the next operation. In '3*10/3" there is no approximation. It is not clear to me how Pioneer models handles the approximation much better. 

06072022, 03:38 PM
(This post was last modified: 06072022 03:45 PM by Didier Lachieze.)
Post: #2




RE: Pioneer vs RPL precision
(06072022 02:26 PM)Marco Polo Wrote: The operation is the classic "(10/3)*3". It depends on the display settings (and rounding): with DISP>FIX 09 or FIX 10 you get 10, but with DISP>FIX 11 or DISP>ALL you get 9.9999999999 as on the RPL models. Btw, this is the same on the HP Prime. 

06072022, 04:09 PM
Post: #3




RE: Pioneer vs RPL precision
(06072022 03:38 PM)Didier Lachieze Wrote:I did not consider this.(06072022 02:26 PM)Marco Polo Wrote: The operation is the classic "(10/3)*3". As a matter of fact all the Pioneer roms i have default on FIX (something) while the RPL models default to ALL. Furthermore, i never noted the peculiar behavior of Pioneer and RPL calculator under such conditions. I would expect, for example with FIX 5, 9.99999 as result 

06072022, 05:12 PM
Post: #4




RE: Pioneer vs RPL precision
(06072022 04:09 PM)Marco Polo Wrote: I would expect, for example with FIX 5, 9.99999 as result The FIX setting does not change the internal value, only the way it is displayed. 9.9999999999 when displayed rounded to 5 decimal places is 10. <0ɸ0> Joe 

06072022, 06:33 PM
Post: #5




RE: Pioneer vs RPL precision
(06072022 05:12 PM)Joe Horn Wrote: The FIX setting does not change the internal value, only the way it is displayed. 9.9999999999 when displayed rounded to 5 decimal places is 10. Now I know what it comes with, but it's not what I want. After all these years I can finally recycle a previous comment. 

06072022, 08:05 PM
Post: #6




RE: Pioneer vs RPL precision
(06072022 06:33 PM)Thomas Klemm Wrote:Exactly what I think about it....(06072022 05:12 PM)Joe Horn Wrote: The FIX setting does not change the internal value, only the way it is displayed. 9.9999999999 when displayed rounded to 5 decimal places is 10.Now I know what it comes with, but it's not what I want. 

06072022, 11:12 PM
Post: #7




RE: Pioneer vs RPL precision
(06072022 08:05 PM)Marco Polo Wrote:(06072022 06:33 PM)Thomas Klemm Wrote: Now I know what it comes with, but it's not what I want.Exactly what I think about it.... Please help me understand what you mean. Are you saying that you want FIX 5 to act like TRUNCATE 5, not ROUND 5? (I'm not trying to be obtuse... I really am not sure what you mean here). <0ɸ0> Joe 

06072022, 11:52 PM
Post: #8




RE: Pioneer vs RPL precision
Dr William Kahan had a lot to say about this subject starting on page 144 of the following oral history.
http://history.siam.org/pdfs2/Kahan_final.pdf "HewlettPackard had come out with a beautifully engineered job called the HP35, which was the first scientific calculator with all the scientific functions instead of just the add, subtract, multiply, divide, and maybe a square root. And then they came out with the HP45, which was an improved version. It had more functionality. But in the meantime, Texas Instruments came out with a calculator that was a great deal cheaper, and here’s how they advertised their calculator. So TI had this advertisement in the papers. It was a fullpage advertisement. It said, “Type in your telephone number. Now,” they said, “Take the logarithm.” The logarithm turns out to be a number form tenpointsomething, or ninepointsomething, actually. “Now hit the exponential key. Do you get your phone number back? You do on our calculator.” HP knew that it was the target of this advertisement because it did that on an HP45, which carried ten digits. You type in your tendigit phone number, take the log, take the exponential, and the last digit or two would change but, apparently, not on the TI calculator. HP was very worried about this, because it seemed to impugn the integrity of their beast. It was a very neat job, the HP35, for all its faults—and it had lots. It was really a very nice job, and then, of course, it went to the HP45, which was just sort of an expanded, extended version of the HP35. And the other guys were getting into the act. What one fool can do, another can, so TI had gotten into the act using relatively similar algorithms.And HP was now embarrassed because it appeared that their calculator was somehow defective, and they were worried about it—I mean, really worried about it. They thought they had a certain reputation, and it was being undermined by this calculator. So fortunately, I asked what the problem was all about, and I said, “Can you send me samples of the calculators for me to play with before I come to the meeting?” And they did. So I had an HP45, and I had an SR51. And I discovered what was happening. It’s true that the HP45’s arithmetic was somewhat grotty in spots, but it wasn’t that bad. But what TI was doing was clever. You see, the 45 did its arithmetic to ten significant decimals, period. Everything was done to ten significant decimals, including the internal algorithms that computer logs and exponentials. TI was doing their arithmetic internally carrying 13 significant decimals, but they only showed you ten. So that meant that, though you type ten digits in, as soon as you did some arithmetic, you had 13 decimal digits. But you only saw ten significant decimals. Well, that could hide a lot of sins, couldn’t it? The TI thing was cheaper, but that’s because HewlettPackard can’t do anything that’s cheap there. Their whole culture is such that, whatever they do, it’s going to be expensive. So I discovered that if you did this log exponential thing seven times, then the last digit would change. You see, their arithmetic at the 13th digit was grottier, if anything could be grottier, than the 45. And because it was worse arithmetic intrinsically, it meant that it didn’t take very long for the error to creep up through those three digits. Seven times was enough. So I then was able to turn up and say, “Look: everybody who looks at that ad is being fooled. They think that the TI machine is reproducing your telephone number, but it isn’t. It’s your telephone number with a last digit diminished by one, followed by a certain number of nines, like two nines and a digit. Then it gets rounded up, you see, so it shows up properly in the display. They round in the display, even though they don’t round the arithmetic.” I said, “You do this seven times, and then you’re going to get something with your digit, less one, and followed by a foursomething something because the arithmetic is so crummy. After you’ve done it seven times, your telephone number changes. Do you feel that that’s honest? Is this an honest ad?” Well, certainly it’s got to be mysterious. Somebody who doesn’t realize what’s going on has to find it mysterious that after he does this seven times, that digit changes. That was a shock, and now they realized that they were in a world that was not the world they thought they were in. Whatever the hell was going on, they really weren’t in control of it, but I also came with a proposal to cure the problem. I said, “You can do what they do, except for one thing: in order to be honest, round every result back to ten digits even if you carry thirteen to compute it.” And I said, “If you do that, then each operation, taken by itself, will give you a rather honest answer, and you can explain this log exponential thing. That’s easy because when you take the log, you’ve got the right log. It’s correct to within just a little bit worse than half a unit in the last digit of the display. Then you can say ‘Now, it’s that error that propagates when you take the exponential because, if we recovered your telephone number, we’d be getting the exponential not of the number that you see before you. It would have to be the exponential of something else.’’’" 

06082022, 05:12 AM
(This post was last modified: 06082022 05:16 AM by Marco Polo.)
Post: #9




RE: Pioneer vs RPL precision
(06072022 11:12 PM)Joe Horn Wrote:Using FIX 5 i expect to see 5 decimals.(06072022 08:05 PM)Marco Polo Wrote: Exactly what I think about it.... The number is rounded for any FIX value, In the specific case the calculators gives a "wrong" information, displaying a rounded (to zero decimals) number instead the expected five decimals. I am not a mathematician nor an expert of calculators principles, so I may be wrong, but as an engineer I find this behavior quite disturbing. Strangely, I never noted it since 1989.... . 

06082022, 05:21 AM
Post: #10




RE: Pioneer vs RPL precision
Have you ever wondered what that SHOW key might be for?


06082022, 06:04 AM
Post: #11




RE: Pioneer vs RPL precision  
06082022, 06:07 AM
Post: #12




RE: Pioneer vs RPL precision
(06072022 11:52 PM)Steve Simpkin Wrote: Dr William Kahan had a lot to say about this subject starting on page 144 of the following oral history.Thank you for the explanation. Things are now clearer. 

06082022, 07:55 AM
Post: #13




RE: Pioneer vs RPL precision
(06082022 05:12 AM)Marco Polo Wrote: Using FIX 5 i expect to see 5 decimals. Just tested on Plus42, set to "ALL" decimals. It displays "10" but SHOW displays 9.999999999...... My understanding, now, is that hystorically the calculator design was aimed to display results in a "confortable" way for most people, hiding were possible the rounding errors and HP choosed the way to "hide" rounding errors only when FIXing the decimals. If my understanding is correct, i can live with it :) 

06082022, 07:46 PM
(This post was last modified: 06082022 07:51 PM by ijabbott.)
Post: #14




RE: Pioneer vs RPL precision
(06082022 05:12 AM)Marco Polo Wrote: Using FIX 5 i expect to see 5 decimals. 3.00000 has five decimal places. Do you think the trailing zeroes don't count as proper digits? The calculator only strips trailing zeros after the decimal point in "ALL" mode. (Yes, I realize I spelt zeros/zeroes two different ways. Both are correct.) — Ian Abbott 

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