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Re: Re: RE:Working Precision

• To: mathgroup at smc.vnet.net
• Subject: [mg24072] Re: [mg24030] Re: [mg23928] RE:Working Precision
• From: David Withoff <withoff at wolfram.com>
• Date: Fri, 23 Jun 2000 02:26:41 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

> Here's an easily defended rule:  Do all arithmetic exactly. When
> you must store the result into a finite sized memory location, 
> round it to the nearest representable number exactly representable
> in that memory location. In case of a tie, round to even.

The problem is that that rule by itself makes no allowance for
keeping track of the accumulated effects of all that rounding.

> On a binary machine with a binary exponent, multiplication by 2 is
> always exact.  Think about it:  it adds one to the exponent, leaving
> the mantissa alone.  There is no error on such a machine operation.

Actually, the absolute error is multipled by 2.  For example,
multiplying 1 +/- 1 by 2 gives (or should give) 2 +/- 2.  It
shouldn't matter what sort of bit-twiddling the machine is
doing internally to arrive at this result.  The mathematics
is the same regardless.

Dave Withoff
Wolfram Research