Euler Identity in Home

[Matt Agajanian]

I'm wondering if this thread's points are making a calculator's usage moot because they're varied in their results and inherent limited precision.

I'm just throwing this out there but, are you saying the calculator has become a moot tool?

Granted, no calculator's has a Cray set of processors but, (forgive me), neither does the WP-34S. On that note, as illustrated in earlier posts, even MS Excel spurts out inaccuracies. By my own tests, in DocumentsToGo (iPhone version), both sin(pi)) and tan(pi()) result in -1.22465E-16 but, cos(pi()) returns -1. By the way, speaking of iPhone apps, the WP-34S app (v1.5) returns -2.38462643383e-16 for both sin(π) and tan(π), where cos(π) returns -1.

[Han]

(04-17-2014 08:29 PM)ColinJDenman Wrote:  I do entirely agree that returning a value is wrong. It should say divide by zero, or not a number or "I'm sorry Dave, I can't do that" or some other indicator. I favour the approach that the calculator tells you that your going near its limitations rather than giving a value that can potentially roll through into further calculation. I liked this example because it is a) obviously wrong and b) too big to justify as a small thing like the 2E-13 stuff.

If you wish, "a calculator's gotta know its limitations". Or I need a shotgun:
http://www.hpmuseum.org/forum/thread-1103.html

Your references are indeed interesting. Thanks.

From the third link is this example for the TI-83.
\[ 5^{14} = 6103515625 \]
\[ Ans/4 = 1525878906 \]
So apparently 5^14 is divisible by 4. Then we check what the calculator is really thinking:
\[ Ans - 1525878906 = .25 \]
Quoting the article:

Quote:Of course, the calculator doesn’t really believe that 5^14 is even as is seen when we subtract the apparent answer from itself. I will leave it to the calculator manufacturers to explain in detail what causes this “lie”. I’ll just say it seems to have something to do with the fact that the actual value of 5^14 ÷ 4 has one more digit than it’s able to display precisely and it won’t resort to scientific notation because 5^14 could be expressed precisely.

[Manolo Sobrino]

(04-17-2014 10:39 PM)Matt Agajanian Wrote:  I'm just throwing this out there but, are you saying the calculator has become a moot tool?

Not exactly, but now that you're asking... I use them for exploring things and personal calculations, they're great for this, and for some exam akin to FE that I'm pondering. But for real work? you had to be out of your mind.

Save for specialised user cases, surveyors or I don't know, pilots, machinists? and maybe a few more, you can't publish things done with a calculator if you want to be taken seriously. It's been a long time since there's much better software out there, better as in standard, reproducible, more accurate and quite often more learned (Excel? not really. It's actually pretty good for many other things, though). And if there's not, you're expected to create it using a good compiler. This is 2014 and people have access to computers.

Keep them for blueprints, but then you have to calculate your final numbers with the real toolchain.

(edit)

Han, that's really a cheap fallacy . What happens there is that the calculator runs out of digit positions to display (all of them have this limitation). It's not telling you that it is divisible by 4, it's just the most accurate result it can show you if you divide by 4. If you want to know if it's divisible by 4, call function mod(result#1, 4) or mod(5^whatever, 4), the answer is, of course, 1... That will break at some point too .

[Han]

My favorite are the sine curves (again, in the third link). Like the division by 4 example, all graphing programs will have issues with screen resolution.

[Matt Agajanian]

(04-18-2014 12:56 AM)Manolo Sobrino Wrote:  

(04-17-2014 10:39 PM)Matt Agajanian Wrote:  I'm just throwing this out there but, are you saying the calculator has become a moot tool?

Not exactly, but now that you're asking... I use them for exploring things and personal calculations, they're great for this, and for some exam akin to FE that I'm pondering. But for real work? you had to be out of your mind.

Save for specialised user cases, surveyors or I don't know, pilots, machinists? and maybe a few more, you can't publish things done with a calculator if you want to be taken seriously. It's been a long time since there's much better software out there, better as in standard, reproducible, more accurate and quite often more learned (Excel? not really. It's actually pretty good for many other things, though). And if there's not, you're expected to create it using a good compiler. This is 2014 and people have access to computers.

Keep them for blueprints, but then you have to calculate your final numbers with the real toolchain.

Let me get a handle on this please. With what your stating, how usable, reliable, relevant, essential and valid were the dozens of application packs for the 67/97 and HP-41 since these calcs were equipped with the mathematical shortcomings we're discussing here?

[Manolo Sobrino]

(04-18-2014 04:40 AM)Matt Agajanian Wrote:  Let me get a handle on this please. With what your stating, how usable, reliable, relevant, essential and valid were the dozens of application packs for the 67/97 and HP-41 since these calcs were equipped with the mathematical shortcomings we're discussing here?

Talking about my field (Physics something), which is the only thing I can say I know something about, and keeping in mind that you'll always find the odd stubborn/creative individual to prove you wrong... they're obsolete since the 80s. I'm sorry Matt. But those are really very good news, it means that we can do several orders of magnitude better.

[Matt Agajanian]

(04-18-2014 05:04 PM)Manolo Sobrino Wrote:  Talking about my field (Physics something), which is the only thing I can say I know something about, and keeping in mind that you'll always find the odd stubborn/creative individual to prove you wrong... they're obsolete since the 80s. I'm sorry Matt. But those are really very good news, it means that we can do several orders of magnitude better.

Thanks. I see your point.

[Eddie W. Shore]

(04-04-2014 12:41 PM)Marcus von Cube Wrote:  

(04-04-2014 05:09 AM)ColinJDenman Wrote:  But e^i*pi in degrees mode gives the 2.07E-13. I begin to feel this is less a question of Exact versus Approximate, but rather some inconsistency internally.

In Home, π is just a floating point (BCD) number, not a symbolic constant. Thus, sin(π) simply isn't zero. I wouldn't be surprised if the result is the correctly rounded result of computing sin(3.14159265359).

Agree. This is nothing new for HP calculators.

[Manolo Sobrino]

(04-20-2014 03:32 PM)Eddie W. Shore Wrote:  

(04-04-2014 12:41 PM)Marcus von Cube Wrote:  In Home, π is just a floating point (BCD) number, not a symbolic constant. Thus, sin(π) simply isn't zero. I wouldn't be surprised if the result is the correctly rounded result of computing sin(3.14159265359).

Agree. This is nothing new for HP calculators.

Yeah, I finally figured out what's going on, it's the CORDIC. It can give you those high precision numbers that have nothing to do with the rest of the other things the calculator does with its working significant digits. It is inconsistent, not bad per se. You gain accuracy in some places and lose it in others, you just have to be perfectly aware of how and where that happens. I think that most of the time that is far from obvious. That's why we argued about rounding and different approaches to this.

[ColinJDenman]

(04-20-2014 05:55 PM)Manolo Sobrino Wrote:  Yeah, I finally figured out what's going on, it's the CORDIC. It can give you those high precision numbers that have nothing to do with the rest of the other things the calculator does with its working significant digits. It is inconsistent, not bad per se. You gain accuracy in some places and lose it in others, you just have to be perfectly aware of how and where that happens. I think that most of the time that is far from obvious. That's why we argued about rounding and different approaches to this.

I hesitate to say, but perhaps this, in the end, is the point I've tried to make. The Prime has a CPU bundle more powerful than the laptop I was using 10-15 years ago. That laptop managed to run Windows, and, grudgingly, MathCAD.

Is the firmware really up to that power -- it doesn't seem remarkably fast nor deliver superior results to preceding HPs. I tend to think the home calculator is a cut and paste job of the scientific calculator HP has used for some years.

The algorithm you refer to was targeted for considerably more primitive hardware of decades ago. Applying it to the Prime is developmentally cost effective, but hardly innovative or making good use of the progress in hardware.

Nor is the decision to ignore the chip's built-in SD card support and other features that could have been made available for a few bucks more. See the Raspberry Pi B for the hardware one could get for a low price. Note especially the 512MB RAM.

HP could do better. It used to be a hallmark that they did, before the beancounters infested the world. Other suppliers such as Casio seem to be willing to make more radical efforts to bring the calculator up to date. Unlike other posters, I think the calculator with a dedicated keyboard and instant availability is still a sensible ergonomic computing niche.

[debrouxl]

Indeed, I think that all of us can agree on the fact that all calculators manufacturers could use newer parts which do more work for about the same amount of energy (Cortex-A5 instead of ARM9), they could provide and enjoy better hardware (the figures of the $150 Nspire CX CAS and Prime calculators wrt. $35 RPi or $45 BBB are pretty damning), they could implement larger mantissas for floating-point numbers (or even arbitrary precision floating-point numbers, at even higher performance cost). But they don't.

Quote:Other suppliers such as Casio seem to be willing to make more radical efforts to bring the calculator up to date.

If you're talking about Casio's newest models such as the fx-CP400 II or the fx-FD10 Pro, I disagree with you

The fx-CP400 is not much of an improvement over the decade-old CP300 and CP330. Sure, it's got a large touch screen, but for the rest... mild processor using uncommon ISA (the Nspire and Prime rightfully use ARM processors, like most of the rest of the modern embedded world), low amount of RAM and Flash (both are far below those of a '2007 Nspire with a grayscale screen), BASIC slowness inherited from the CP300/330 (a simple high school level algorithm was shown to execute two orders of magnitude slower on a fx-CP400 than on a TI-82), initial OS versions not taking advantage of color and touch as much as they could, closed platform - and last but not least, a higher price tag than a Nspire CX CAS or a Prime...

The fx-FD10 Pro's point is to be ruggedized for midly harsh environments, and indeed, it should do (much) better at that task than other calculator models. Beyond that, well... the "superior brain" claimed on Casio's page is actually the same old brain as other existing Casio models, harking back even further in the past. Less user-accessible RAM than a '1995 TI-92, less user-accessible Flash memory than a '2002 TI-V200 (hardly more than a '1998 TI-89 or '2003 HP-49G+). There's a SD card slot, but the HP-49G+ has provided that for over a decade.

[Manolo Sobrino]

(04-27-2014 10:59 PM)ColinJDenman Wrote:  I hesitate to say, but perhaps this, in the end, is the point I've tried to make. The Prime has a CPU bundle more powerful than the laptop I was using 10-15 years ago. That laptop managed to run Windows, and, grudgingly, MathCAD.

Is the firmware really up to that power -- it doesn't seem remarkably fast nor deliver superior results to preceding HPs. I tend to think the home calculator is a cut and paste job of the scientific calculator HP has used for some years.

The algorithm you refer to was targeted for considerably more primitive hardware of decades ago. Applying it to the Prime is developmentally cost effective, but hardly innovative or making good use of the progress in hardware.

Nor is the decision to ignore the chip's built-in SD card support and other features that could have been made available for a few bucks more. See the Raspberry Pi B for the hardware one could get for a low price. Note especially the 512MB RAM.

HP could do better. It used to be a hallmark that they did, before the beancounters infested the world. Other suppliers such as Casio seem to be willing to make more radical efforts to bring the calculator up to date. Unlike other posters, I think the calculator with a dedicated keyboard and instant availability is still a sensible ergonomic computing niche.

I was talking about the Saturn calculators. And it gets funnier, because I suspect (with good reasons) that the way they could get the CORDIC to do that was by storing double precision values in the algorithm tables. That is, using 24 significant digits of Pi internally. At least, that's what TI did with the 68 to improve range reduction, and hence it can give you very accurate results for sin(integer*pi`14) that match those of the HPs... then they went back to using only single precision values of Pi and that's why they now get zero and have worse range reduction. BTW, the HPs have excellent range reduction . You can't trust calculator designers to tell you everything

I don't believe this is a straight cut & paste job, because that would mean that they're emulating Kinpo emulating Saturn... I don't think so. I guess they're recreating an ideal HP 12 digit calculator (that nevertheless works under the hood effectively with 24 to give you whatever 12 it couldn't otherwise). Why not 14 or 16? I don't know, maybe Saturn's 12 is a magic number. They're not using guard digits because they believe that the rest of the people who work with these things are wrong and blah blah blah.

I fully agree with you. I passed my Numerical Methods and got to grips with Mathematica 2.2 in the middle '90s with a 26 MFLOPS Pentium 90 with 8+32MB of RAM (an awfully expensive HP Vectra BTW). Even though apparently this ARM is from 2004, it still claims 215 MFLOPS (pinch of salt there.) But alas, management, resources allocation, fans... This is just some company's product after all, not an unquestionable encapsulation of mathematical wisdom, it really "really" doesn't matter too much.

(edit)

And it gets even more funny! Look at this fine piece of misleading calculator hype, now from the other side:
Solution 10702: Unexpected Results When Sin(Pi) Does Not Return Zero on the TI-68. Sounds familiar... Well, they changed their mind later and at least they're consistent now. Any company can sell you what you're willing to buy.