Re: Standard Library Interest?

[Dmitry A. Kazakov <mailbox@dmitry-kazakov.de>]

On Wed, 08 Oct 2003 21:46:52 GMT, "Robert I. Eachus"
<rieachus@comcast.net> wrote:

>Marin David Condic wrote:
>> It always has been and always will be a "Political Problem". We've known 
>> for aeons how to make a square root routine. Getting one that could be 
>> accepted as "Standard" was a political problem. Just because its 
>> "political" doesn't make it any less a "problem", so we need to find the 
>> "political" will to do it. From there, the technology is relatively simple.
>
>I don't want to harp on this, but WHO has known how to make a square 
>root routine for aeons?  In the early sixties it was vital to NASA to do 
>  square roots fast and accurately for the Apollo project.  I worked on 
>this, and the eventual result was to modify four IBM 7094s to include a 
>36-bit to 36-bit integer square-root function.  I won't go into a long 
>discussion of WHY a 36-bit to 36-bit square-root with the implicit 
>binary point to the right of the most significant digit was what was 
>needed.  Just the thought of explaining to ONE more person why that was 
>what was needed, and that it was a very simple modification of the IBM 
>7094 to add it makes me want to scream.
>
>In the mid-eighties when I was at Stratus, I showed the same thing to 
>the hardware engineers responsible for the math co-processor software in 
>the Stratus 200 and 600 series.  They agreed it would significantly 
>improve the speed of their square root routines, but AFAIK, it was never 
>implemented.  (This was when we were about to introduce the 2000 series 
>which used Motorola 68881 math co-processors.)  Later, I implemented 
>this square-root method in software for Sun SPARCs, since it was so much 
>faster than Sun's math routines for most SPARC chips.  (Most UltraSPARC 
>chips now seem to do the square-root correctly.)
>
>So yes, I know how to use the Newton-Rhapson method to find square 
>roots, and I also know how to implement a square-root in microcode so 
>that it is faster than the hardware divide.  But the idea that this was 
>well or widely known in 1983 is totally false.  There were CPUs in the 
>nineties that still did it wrong.  (Of course, in the nineties Intel 
>managed to get divide wrong as well.  Sigh!)  Most implementations of 
>sine and cosine were pretty good, but other trig functions and all 
>inverses were spotty.  Logs again were spotty both natural and common 
>logs, and most implementations of exp were a horror.  Brian Wichmann did 
>a lot of work on that, including finding cases where many compilers 
>computed X**4 wrong, even when X was integer and the result was 
>therefore exact.  (The IBM 360 series and follow-ons were infamous for 
>printing the sum of 2.0 and 2.0 as 3.9999999999999...  But that was 
>actually a problem in the Fortran print routines, not in the internal 
>representation.)

Hear! Hear!

I remember, an excellent book on numerical methods (Forsyte's, I
believe) in early 80s, which started with a square root equation and
showed how difficult in fact to solve such an "elementary" thing
properly. That time there was no numerical library which had a decent
subprogram for that.

---
Regards,
Dmitry Kazakov
www.dmitry-kazakov.de