Changing Of The Guard

[Matt Agajanian]

Hi all.

If I recall correctly, Spices/Spikes and prior only maintained 10-digit accuracy with no guard digits. So, correct me if I'm wrong. What are the calculation/accuracy forensics for each lineage following?

Thanks

[Thomas Klemm]

(04-21-2014 09:51 PM)Matt Agajanian Wrote:  So, correct me if I'm wrong.

What's the expected result of 1.000050005 * 1.00001 with and without guard digits?
The exact result is 1.00006000550005.
I get 1.000060006 with the HP-35.

HTH
Thomas

[bshoring]

I believe the first HP with guard digits was the HP-91 printing scientific, with 3 guard digits. It would make sense to me that any calcs that came out after it, including the late Woodstock, and all that followed would likely have those guard digits.

Hopefully someone can verify if that's the case. As best I can tell, Spice, Nut and Voyager calcs had about the same level of accuracy.

Bob

[Mike Morrow]

(04-22-2014 01:50 AM)bshoring Wrote:  I believe the first HP with guard digits was the HP-91 printing scientific...

What an odd belief. :-)

That certainly is not true about the HP-91 nor none of the other machines that you mention...nor of the HP-67/-97, nor the HP-41C series, nor the HP-15C...etc. etc. etc. AFAIK, no HP calculator used any guard digits until machines based on the Saturn processor were developed. That would be around 1986...the HP-28C and later. Even then...the additional digits are strictly internal to the specific calculation and do not survive when the number reaches the display. For example, the HP-42S is said to use 15-digit precision during calculation. But for a simple example. pushing the PI button produces a display of 3.14159265359. Take the SIN of that, and the exact same result is produced that occurs when you key in 3.14159265359 and take the SIN. There are no hidden digits in the value that is displayed after PI is executed. The same is true for more complex examples.

TI machines used three guard digits at least as early as the SR-50 in 1974. Those non-displayed digits still exist when the value is presented for display. Subtracting a manually keyed-in value that appears identical to the displayed value will generally show the presence of the guard digits in the original value. Most Casio machines (like the fx-115ES Plus) show the same behavior, except that the 115ES has five guard digits! I personally always preferred TI's use of guard digits to HP's refusal to improve the precision of calculations on their machines. HP always bizarrely maintained that their failure to provide guard digits was a positive characteristic, compared to TI. That was pure HP BS. For example, I always got far better accuracy of results, due to far better precision, from a fourth-order Runge-Kutta program on a TI-59 than I ever got from the same program executed on a HP-67 or HP-97 or 41C. This is just another of many areas where HP screwed the pooch in its mythological age of excellence. :-)

[Thomas Klemm]

(04-22-2014 01:50 AM)bshoring Wrote:  Hopefully someone can verify if that's the case.

The HP-35 already had 2 guard digits. You can verify this by calculating 1.0005005². Actually all 13 digits are calculated. But then the last digit is lost by shifting the result to the right. This was corrected later.

Cheers
Thomas

[Matt Agajanian]

Mike, Yes, indeed. Even as I remember from my SR-56 days, TI would always reprint that guard digits/ EE/INV-EE guard digits example even in its TI-58/59 and TI-57 manuals among others to show that guard digits were present in calculations.

Thomas, thanks for the verification. Although, as I regard HP in high technological respect, their usage of guard digits in their calcs is quite obscure and I do not recall in any manuals that calculations are carried past 10 digits of accuracy (unless it's explicitely stated in some of their manuals and I must've missed the fine print somwhere).

[Thomas Klemm]

(04-22-2014 05:04 AM)Mike Morrow Wrote:  AFAIK, no HP calculator used any guard digits until machines based on the Saturn processor were developed.

These are intermediate results of calculating 1111111*1111111 = 1234567654321.

HP-35
Before rounding:


After rounding:

Note: the last digit is lost.

HP-34C

Note: the rounded result is in register c.

Cheers
Thomas

[Thomas Klemm]

(04-22-2014 05:28 AM)Matt Agajanian Wrote:  Thomas, thanks for the verification. Although, as I regard HP in high technological respect, their usage of guard digits in their calcs is quite obscure and I do not recall in any manuals that calculations are carried past 10 digits of accuracy (unless it's explicitely stated in some of their manuals and I must've missed the fine print somwhere).

In HP calculators the guard digits are not exposed to the user. The intermediate result is rounded to 10 or 12 digits. You always get what you see. TI calculators only displayed the rounded result while keeping the guard digits internally. This is an advantage if you chain computations but can be confusing because e.g. you see a 2 but internally it is 1.9999999999. Start squaring this number to see what can happen.

Cheers
Thomas

[Joe Horn]

(04-22-2014 05:28 AM)Matt Agajanian Wrote:  Although, as I regard HP in high technological respect, their usage of guard digits in their calcs is quite obscure and I do not recall in any manuals that calculations are carried past 10 digits of accuracy (unless it's explicitely stated in some of their manuals and I must've missed the fine print somwhere).

It's explained in at least some HP manuals. For example, the 32SII says on page 1-16, "All numbers are stored with 12-digit precision.... During some complicated internal calculations, the calculator uses 15-digit precision for intermediate results."

Ever since Dennis Harm's famous "The New Accuracy: Making 2^3=8" article was written (HP Journal, October 1976, pages 16-17), every HP BCD calculator has used 3 extra digits during internal calculations, and at least some of their manuals have in fact explained this.

[DavidM]

(04-25-2014 10:44 AM)Joe Horn Wrote:  Ever since Dennis Harm's famous "The New Accuracy: Making 2^3=8" article was written (HP Journal, October 1976, pages 16-17), every HP BCD calculator has used 3 extra digits during internal calculations, and at least some of their manuals have in fact explained this.

That's a nice reference article, Joe. It may have also come in handy during another recent discussion where trig functions and \(\pi\) were being discussed to show the rationale behind some of the non-CAS results (even if they weren't agreeable). You have an amazing amount of historical info stored away in your head! Thanks for sharing it with us.

[jte]

’Mike Morrow’ pid=’9614’ dateline=’1398 Wrote:⋮
⋮ lots of good stuff snipped…
⋮
TI machines used three guard digits at least as early as the SR-50 in 1974. Those non-displayed digits still exist when the value is presented for display. Subtracting a manually keyed-in value that appears identical to the displayed value will generally show the presence of the guard digits in the original value.

Last month I was (in connection with a bug ticket filed for the HP Prime) writing an email to a teacher about hidden guard digits and why the HP Prime differs from other calculators commonly used in classrooms when punching in 1/3*3. I thought I’d use a very simple calculation with a TI-84 to reveal its hidden guard digits — simply 1/3*3-1. But that didn’t reveal that the apparent “1” from 1/3*3 wasn’t actually 1 (“1/3*3-1” had “0” showing up on the display).

So I wholeheartedly agree with the “generally” above! (I.e., not always!) (Using the TI’s floor function revealed that 1/3*3 wasn’t actually 1, as did raising the apparent “1” to the power of 10^10.)

Quote:Most Casio machines (like the fx-115ES Plus) show the same behavior, except that the 115ES has five guard digits! I personally always preferred TI’s use of guard digits to HP’s refusal to improve the precision of calculations on their machines. HP always bizarrely maintained that their failure to provide guard digits was a positive characteristic, compared to TI. That was pure HP BS. For example, I always got far better accuracy of results, due to far better precision, from a fourth-order Runge-Kutta program on a TI-59 than I ever got from the same program executed on a HP-67 or HP-97 or 41C.

To me, at least theoretically, hidden guard digits and precision are two distinct choices. (“Theoretically” as practical computation limits may well have the two decisions connected.) From a debugging / understanding perspective, it is nice to be able to easily see the exact state of a machine. Perhaps a compromise is possible: have the default display mode show less than full precision (hide digits) yet let the user get the calculator to display the numbers as actually stored by changing a setting.

[teenix]

(04-22-2014 05:04 AM)Mike Morrow Wrote:  

(04-22-2014 01:50 AM)bshoring Wrote:  I believe the first HP with guard digits was the HP-91 printing scientific...

What an odd belief. :-)

AFAIK, no HP calculator used any guard digits until machines based on the Saturn processor were developed.

HP-9100A, 2 guard digits - Page 12 of user manual.

cheers

Tony