> -----Original Message----- > From: tclug-list-bounces at mn-linux.org > [mailto:tclug-list-bounces at mn-linux.org]On Behalf Of Mike Miller > Sent: Thursday, July 20, 2006 11:06 AM > > He was only saying that 8-digits is enough for "slide rule > replacement", > which makes sense because if a slide rule can do three digits, then a > device that can do 8 digits is more than sufficient to > replace a slide > rule. Thanks. 8 digits is more than adequate for slide rule replacement, but inadequate for optical design and for some electronic development tasks. > I would like to hear from Chuck what he was doing that > required 10 digits > instead of 8 digits. That's pretty interesting. I won't go into detail, but will suggest a clue to understanding why optical design (lens design) may need extra high precision. Note that a color-corrected lens would be expected to show the hair on a critter that is more than 17 feet away. The angular size of a hair at 17 feet is an arc-second. The calculations for surfaces and ray bending through a 2 or more element lens must be accurate to arc seconds over a field of view of say 45 degrees. That's too wide for small-angle approximations and 8 digits are significant, so more are required to assure there is no round-off error in thousands of iterations. A computer is generally a better choice for any optical design, but not always and not for everybody, etc. The case that made me study this long ago was in developing a pulsed laser rangefinder to do accurate survey rangefinding of 1/10 inch accuracy over distances of up to two miles. Before building the arithmetic unit for display to people, I had to check and prove accuracy in the field as part of debugging the analog radar-like "front end". The front end converted pulse transit time samples to a digital count. That required correction for the atmospheric effects upon the speed of light as well as scale factors to get from a machine count to inches. For reasons I don't recall, that took at least 8 digits without any round-off at all... I think it was actually ten, but the reason for that extra bit isn't obvious from what I just outlined... Oh! that machine count was an accumulated machine average of 40,000 samples for noise reduction, and the least digits were significant. > I'd also > wonder about > internal precision because I believe my old calculator that showed 8 > digits actually had a little more inside and you could see > those extra > digits by subtracting away the 8 visible digits. HP had very good specs and app notes that detailed the numeric precision of their analytic and transcendental calculations. TI's notes weren't as good, but clearly showed less precision and its inability to do the more precise tasks. I don't remember exact details and won't look them up now, but I think you are right but not complete. I recall that there was more analytic function accuracy than transcendental in these. Transcendentals accurate to only 1 part in 10exp8 or less. When carried through 10-100 iterations will make the errors limit a result's repeatability to only one part in 10exp6 or less. The extra analytic (exact function) digits "helped" in complex calculations. A classic error analysis of a mixed calculation would make all this clear. I separately built some digital numerical machines to do analog computer stuff and had studied the behavior of analytic functions versus transcendentals: that confirmed that some forms of recursion on analytic functions have no cumulative error while that cannot be true for transcendentals. If you're really interested, Google some old HP info on calculation precision and accuracy. That should be available, and will cover the HP calculator precision and accuracy capabilities quite well. Might have it in my old HP calculator books. Chuck