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

[Dave Wade]

> -----Original Message-----
> From: cctalk <cctalk-bounces at classiccmp.org> On Behalf Of Jon Elson via
> cctalk
> Sent: 09 January 2019 17:43
> To: Paul Koning <paulkoning at comcast.net>; General at ezwind.net;
> Discussion at ezwind.net:On-Topic and Off-Topic Posts
> <cctalk at classiccmp.org>
> Subject: Re: Teaching Approximations (was Re: Microcode, which is a no-go
> for
> 
> On 01/09/2019 07:49 AM, Paul Koning via cctalk wrote:
> >
> > 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
> A real problem on the IBM 360 and 370 was their floating point scheme.
They
> saved exponent bits by making the exponent a power of 16, instead of 2.
> This meant that the result of any calculation could end up normalized with
up
> to
> 3 most-significant zeros.  That would reduce the precision of the number
by
> up to 3 bits, or a factor of 8.  Some iterative solutions compared small
> differences in successive calculations to decide when they had converged
> sufficiently to stop.
> These could either stop early, or run on for a long time trying to reach
> convergence.
> 

Early machines had to have the floating point units re-worked as IBM found
they had to add guard bits to the calculations to get any thing like decent
accuracy. In general you needed to use double for decent results.


> IBM eventually had to offer IEEE floating point format on later machines.
> 
> Jon

Dave