SR-50: A bumpy road??

[Matt Agajanian]

Hi.

We all know about HP-35’s ln 2.02 and trig function inaccuracies of arguments close to 90⁰. Yeah, yeah. TI probably learned from HP’s hiccups. What were the stories about the SR-50’s mathematical pitfalls?

[Steve Simpkin]

William Kahan had the following to say about the apparent "increased accuracy" of TI models when he worked as a consultant for HP.

"Hewlett-Packard had come out with a beautifully engineered job called the HP-35, 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 HP-45, 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 full-page advertisement. It said, “Type in your telephone number. Now,” they said, “Take the logarithm.” The logarithm turns out to be a number form ten-point-something, or nine-point something, 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 HP-45, which carried ten digits. You type in your ten-digit 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 HP-35, for all its faults—and it had lots. It was really a very nice job, and then, of course, it went to the HP-45, which was just sort of an expanded, extended version of the HP-35. 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 HP-45, and I had an SR51. And I discovered what was happening. It’s true that the HP-45’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 Hewlett-Packard 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 four-something 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."

This is discussed starting around page 144 on the following interview with Dr. William Kahan - August 2005
https://drive.google.com/file/d/1Jlg9EWQ...zwcol/edit

[Matt Agajanian]

And yes, I looked at Joerg’s Datamath page about Calculator Forensics. Still, other dissertations and articles would be interesting.

[Maximilian Hohmann]

Hello!

(06-13-2024 03:33 AM)Steve Simpkin Wrote:  William Kahan had the following to say about the apparent "increased accuracy" of TI models when he worked as a consultant for HP.

This text has been quoted on this forum many times already. I still don't think that it is worthy of someone like Dr. Kahan. It is trivial knowledge that digital machines with limited resources are limited to a certain number of decimal places. In this case, Ti got it right by computing more decimal places than can be displayed, so for simple calciulations and for complicated calculations repeated several times, all displayed digits are correct. HP got it wrong because the number of digits in the display is the same as the internal precision. There is nothing to see here and nothing to discuss and of course any computer will display wrong digits if you only repeat a calculation often enough. One time for HP and seven times for Ti in this case.

Regards
Max

[Matt Agajanian]

HP’s accuracy vs display is what boggles my mind. Even though HP was quite a reputable company back then (no reflection on their current integrity), why weren’t they aware to ensure display accuracy extra unseen digits would validate that the results seen are reliable.

Yes, yes. More accuracy means more processing, more ROM, lengthier algorithms, computation time meant higher manufacturing and product costs. Maybe HP could have made a $395 product with higher accuracy work out. In any case, why were the accuracy reliabilities and their processes abandoned?

[brouhaha]

The HP-35 was to a large extent meant to be an improved replacement for slide rules, meaning that it should ideally have at least five significant digits. Each additional digit cost money (as you yourself poined out), and they chose a ten digit significand (often incorrectly called mantissa), even though they knew there would be limitations on the accuracy of transcendental functions. If they had chosen twelve digits, people would be asking why they didn't choose fourteen. No matter what a product has, there are always people that want more. At some point a decision has to be made and stuck with, and they chose ten, which was an enormous improvement over a slide rule.

Once they chose a ten digit significand (fourteen digit word), it would have cost a LOT to change that for later calculators, so they only changed it twice in HP-engineered calculators up until the 1990s:
* shortened to eight digit significand for HP-01 watch (twelve-digit word)
* lengthened to twelve digit significand (sixteen-digit word) for Saturn (all new models from 1984 forward)

Quote:In any case, why were the accuracy reliabilities and their processes abandoned?

They weren't!

Despite sticking to the ten-digit significand for stored numbers for all calculators through 1983 (except the HP-01), they didn't rest upon their laurels and keep the transcendental limitations of the HP-35. They made numerous improvements that substantially increased the accuracy, with almost every successive generation of calculators. The HP-35 had such limited ROM that the internal constants used in the transcendental had lower precision, and that was upgraded in the HP-45, and upgraded more in the Topcat and late Woodstock series. They also went to using a thirteen digit significand for internal computations, and only rounded down to ten digits for storage and display.

Some of the improvements were described in HP Journal articles by Dennis Harms (The New Accuracy: Making 2³= 8) and William Egbert (Personal Calculator Algorithms, parts I through IV).

Quote:Maybe HP could have made a $395 product with higher accuracy work out.

They did! Just not in 1972.

Given how tremendously successful HP calculators were for many years, I personally wouldn't dream of telling them they chose the wrong number of digits. There is no evidence to support the idea that they would have been more successful or profitable with more digits. Remember, their job was to make money, not to please enthusiasts (except to the extent that doing so improves profitability).

I'm not claiming that HP was perfect, or even that they did the best possible job under the circumstances. Even with the benefit of hindsight, that's impossible to know. I'm just saying that they got a LOT right, and that we armchair quarterbacks even with half a century of additional market knowledge wouldn't necessarily do any better job under the same constraints, and could easily do worse.

If I were going to criticize any aspects of 1970s HP calculator engineering, it would be:
* the critical dependency on the NiCd battery (and good electrical contact with same) to limit the DC voltage in the Woodstock calculators to prevent damaging the MOS chips
* the press-fit assembly of Spice calculators (which they later eliminated, reverting to soldered assembly)

[Maximilian Hohmann]

Hello!

(06-14-2024 08:25 AM)brouhaha Wrote:  I'm not claiming that HP was perfect, or even that they did the best possible job under the circumstances.

This is certainly correct. But the competition, in this case Ti, exploits the tiniest little weakness to their benefit... As if the significantly lower cost of the Ti SR-50 would not have been enough to convince the buyers. By the way: Comparative advertising was not allowed in my part of the world in these days so we did never see adverts like the one from Texas Instruments.

One year later (1974) Sinclair in the UK came out with the "Sinclair Scientific" that sold in crazy numbers even if it offered just three (!) digits of precision with transcendental functions. People just didn't care because they no longer had to look up the numbers in tables and those wouldn't give them more digits either. And really: nobody, really nobody, ever needed 10 digit precision in any calculation that was ever done on a pocket calculator.

But still to this day, Swiss Micros boasts with 34 digits of precision when it comes to advertising the most expensive scientific calculator currently on the market... so numbers obviously sell! Just as with "top speed" which helps to sell cars in countries with strict speed limits like Switzerland :-)

Regards
Max

[Johnh]

There's a point at which lack of calculator accuracy really does impact our confidence about the result, and I had that Sinclair Scientific from 1974. I think I got it for 20 pounds, in 1975. It was neat and fun, but not actually much use, even at high school. Those trig functions, took about 15 seconds to get 3 figures, and I could read four-figure tables quicker than that. But by 1977 I had a Commodore SR1800, made in 1976 that had all the basic scientific functions needed at 12 digits internal precision, and a very functional ,reliable and economucal build. A massive development in accessibility and affordability since the hp35 lead the way just 3 years earlier.