Pioneer vs RPL precision

[Steve Simpkin]

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

"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-pointsomething,
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.’’’"