Teaching Approximations (was Re: Microcode, which is a no-go for

[dwight]

Recently, in the calculator group, a fellow has an older calculator and was using the power function. You know x^y. He'd put in 2^2 and got 3.99998 or something where the last digit was 2 off. He was worried that the calculator was bad. I explained to him that the calculator was fine. It was simply a rounding error. I explained that the calculator was not doing 2*2 and that it used some series or such to calculate the general case. Most calculators today have about 2 extra bits not shown to hide most rounding errors. Early calculators didn't have any extra digits.
The Intel case was actually worse as its errors were often large and not just in the last digits, when they happened. The errors tended to cluster around odd integer values as well. When looking at all the possible numbers, the odds were small. When looking at integer values the odds were a lot larger and the errors were also larger.
Dwight


________________________________
From: cctalk <cctalk-bounces at classiccmp.org> on behalf of Paul Koning via cctalk <cctalk at classiccmp.org>
Sent: Wednesday, January 9, 2019 5:49 AM
To: Tony Duell; General Discussion: On-Topic and Off-Topic Posts
Subject: Re: Teaching Approximations (was Re: Microcode, which is a no-go for



> On Jan 8, 2019, at 11:58 PM, Tony Duell via cctalk <cctalk at classiccmp.org> wrote:
>
> ...
> IIRC one of the manuals for the HP15C had a chapter on 'Why this
> calculator gives the wrong answers'. It covered things like rounding
> errors.
>
> -tony

That reminds me of a nice old quote.

"An electronic pocket calculator when used by a person unconversant with it will most probably give a wrong answer in 9 decimal places" -- Dr. Anand Prakash, 9 May 1975

Understanding rounding errors is perhaps the most significant part of "numerical methods", a subdivision of computer science not as widely known as it should be.  I remember learning of the work of a scientist at DEC whose work was all about this: making the DEC math libraries not only efficient but accurate to the last bit.  Apparently this isn't anywhere near as common as it should be.  And I wonder how many computer models are used for answering important questions where the answers are significantly affected by numerical errors.  Do the authors of those models know about these considerations?  Maybe.  Do the users of those models know?  Probably not.

        paul