Dividing factorials on 15C CE

[Steve Simpkin]

(01-10-2024 09:22 AM)Johnh Wrote:  

Quote:I think this also goes with the HP traditional philosophy of showing you the computed answer based on the number of significant digits stored by the hardware without using invisible "guard-digits" to round off the answer as TI and other calculator manufactures typically do.

There is some interesting history behind the decision on what to show the calculator user when the true result using fixed digit math is not what they expect to see. William Kahan had the following to say about the apparent "increased accuracy" of TI models when he worked as a consultant for HP.
This is discussed starting around page 144 on the following interview with Dr. William Kahan - August 2005
https://drive.google.com/file/d/1Jlg9EWQ...zwcol/edit

Thanks, that's an interesting read. Kahan talks engagingly about his work, which he was rightly proud of.

But I'm curious about the points about rounding and guard digits. Seems to suggest that HP round every result step to 10 sig figs, rather than carrying the guard digits over to the next step. (screen shot from page 146 attached. )

I can see how this would address the TI example of doing multiple repeats of e^(ln(telephone number)). But I'm not following why it's a better idea for real-world calcs where results need to be carried over? It seems to me that over several different calc steps you would probably loose accuracy in the 10th and maybe 9th digit, instead of probably keeping them accurate if 13 digits are maintained internally?

HP calculators did not really use guard digits. The memory and registers on the pre-Saturn CPUs were 56-bits in size which stored a 10 digit mantissa, a 2 digit exponent and signs for the mantissa and exponent. This is also what was displayed. It may have been possible to internally calculate more than 10-digits since the algorithms are working serially on 4-bit BCD digits at a time but it could only store 10-digits of the result.

The TI models used 64-bit registers and could typically store a 12 or 13 digit mantissa while displaying only 10 digits (2-3 guard digits).

There is an article by Dennis Harms ("The New Accuracy: Making 2 to the third = 8") in the Nov 1976 issue of HP Journal magazine that discusses how HP implemented the improved accuracy William Kahan suggested. A PDF of this issue is available here:
https://www.hpl.hp.com/hpjournal/pdfs/Is...976-11.pdf