Accuracy Management of early HPs

[Steve Simpkin]

(09-10-2022 07:22 PM)Matt Agajanian Wrote:  SNIP

(09-10-2022 04:49 AM)Steve Simpkin Wrote:  Dr William Kahan was directly responsible for this increased accuracy, starting with the HP-27. See page 144 of the following oral history.
http://history.siam.org/pdfs2/Kahan_final.pdf

I said, “You can do what they do, except for one thing: in order to
be honest, round every result back to ten digits even if you carry thirteen to compute it.” And I
said, “If you do that, then each operation, taken by itself, will give you a rather honest answer,
and you can explain this log exponential thing. That’s easy because when you take the log,
you’ve got the right log. It’s correct to within just a little bit worse than half a unit in the last
digit of the display. Then you can say ‘Now, it’s that error that propagates when you take the
exponential because, if we recovered your telephone number, we’d be getting the exponential not
of the number that you see before you. It would have to be the exponential of something else.’’’

SNIP

Clarify some things, please.

1--Am I to understand that to preserve and display accurate results

(Step 1) a calculation is carried out to 13 places.

(Step 2) this calculated result is displayed and rounded to 10 digits (although 13 digits are internally maintained)

(Step 3) the next calculation uses the 10 digit displayed figure, but calculated with the internal 13 digit result.

(Step 4) Go back to Step 2.

Or am I missing something?

I may not have a complete understanding of what is going on internally but I believe the sequence is more like the following:

(Step 1) An "intermediate" calculation is carried out to 13 places. I believe this can be done even though the HP hardware only allows the storage of 10-digit numbers since the algorithms are working on a 4-bit BCD digit at a time.

(Step 2) This calculated result is rounded to 10 digits and is stored as a 10-digit number. This number is also displayed as a 10-digit number. Keep in mind the the HP hardware has 56-bit data registers that can only store 10-digit BCD numbers (10 digit mantissa, a 2 digit exponent and signs for the mantissa and exponent).

(Step 3) The next calculation uses the 10 digit stored numbers, since that is all that is available with 56-bit registers.

(Step 4) Go back to Step 1.

I Believe the TI calculators used more than 56-bits in their data registers so they could actually store the 13 (or higher) digit results of calculations, even if they only displayed 10 (or less) rounded digits of that register.