(12C Platinum) Internal Precision Test
12-31-2018, 10:33 AM
Post: #1
 Gamo Senior Member Posts: 497 Joined: Dec 2016
(12C Platinum) Internal Precision Test
This well know program shown that 1/n! can be used to verify the series
e = 1/0! + 1/1! +1/2! + ... + 1/n! to approximates the value of e .

Each iteration compare this to the 12C Platinum's value of e .

-----------------------------------------

Procedure:

[R/S] display each iterations from this formula Pause and display calculator's e
When both are equal then display the total iterations.
[X<>Y] to view the value of e

-----------------------------------------

12C Platinum took 14 iteration which mean that the internal precision is more than the Original 12C where 12C took 11 iterations.

Program: ALG Mode
Code:
 0 STO 1 STO 0 RCL 0 n! 1/x + RCL 1 =  FIX 9 STO 1 PSE 1 e^x PSE X≤Y GTO 021 1 STO+ 0 GTO 004 RCL 1 RCL 0

Gamo
12-31-2018, 01:52 PM (This post was last modified: 12-31-2018 01:59 PM by Albert Chan.)
Post: #2
 Albert Chan Senior Member Posts: 633 Joined: Jul 2018
RE: (12C Platinum) Internal Precision Test
(12-31-2018 10:33 AM)Gamo Wrote:  12C Platinum took 14 iteration which mean that the internal precision is more than the Original 12C where 12C took 11 iterations.

IIRC, 12C internal precision digits are rounded-away after each operation.
Only 10 sig. digits remains (what you see is what you get).

If the same applied to 12C Platinum, why did it need more terms to reach 10 digits of e ?
I would guess it would need about the same iterations:

2
2.5
2.666666667
2.708333334
2.716666667

2.718055556
2.718253969
2.718278771
2.718281527
2.718281803

2.718281828

BTW, getting it converge to 10-digits of e is just lucky. Last digit might be off due to rounding error.
Example, next iteration of above is 2.71828183

If the goal is to test internal precision of calculator, this test is not very good.
12-31-2018, 02:21 PM (This post was last modified: 12-31-2018 03:13 PM by Dieter.)
Post: #3
 Dieter Senior Member Posts: 2,398 Joined: Dec 2013
RE: (12C Platinum) Internal Precision Test
(12-31-2018 01:52 PM)Albert Chan Wrote:  IIRC, 12C internal precision digits are rounded-away after each operation.
Only 10 sig. digits remains (what you see is what you get).

No, the 12C Platinum does not finally round to 10 digits. We have verified this in an earlier thread.

Also let's be accurate about the term "internal precision". This usually referns to the precision of the calculator's internal (sic!) calculations. On HP calculators these are typically three more digits than what is exposed to the user, i.e. 13 dgits for 10-digit calculators or 15 digits for 12-digit devices.

What we are talking about here is something different. The 12C Platinums works with 12 digits while the hardware can only display 10 of these. In a way that's similar to, say, the TI-58/59: you see 10 digits but 13 are present.

So while the 12C Platinum displays √2 as 1,414213562 there calculated result is 1,41421356237. Which can be shown by subtracting the displayed value, this should yield 3,7E–10.

So...

(12-31-2018 01:52 PM)Albert Chan Wrote:  2
2.5
2.666666667
2.708333334

...I am pretty sure the 12C Platinum wil show 2,708333333 at this point, avoiding the 10-digit roundoff error.

(12-31-2018 01:52 PM)Albert Chan Wrote:  If the same applied to 12C Platinum, why did it need more terms to reach 10 digits of e ?
I would guess it would need about the same iterations:

The additional iterations occur because at the end there are two values that look the same (the first 10 digits agree) while the test in the program detects that they are still different, so the iteration continues until they match in all 12 digits.

Since I don't have a 12C Platinum at hand I have tried this on the 12-digit HP35s. Set to 10 displayed digits, the result eventually reaches 2,718281828. But the iteration does not stop here because the 12-digit value at this point is 2,71828182829 which is compared to the 12-digit true value of e, 2,71828182846. The next iteration yields 2,71828182845 and finally 2,71828182846. These are the mentioned two additional iterations. Gamo said it's three more iterations on the Platinum, but this is caused by roundoff errors in the standard 12C: The last value in your table actually should be 2,718281826 when evaluated exactly, which means one more iteration. It should be 12 and 14 instead of 11 and 14.

(12-31-2018 01:52 PM)Albert Chan Wrote:  BTW, getting it converge to 10-digits of e is just lucky. Last digit might be off due to rounding error.
Example, next iteration of above is 2.71828183

Yes, with 10 digit precision the iteration converges to 2,718281830. That's also caused by the mentioned roundoff errors.

Dieter
12-31-2018, 03:32 PM
Post: #4
 Albert Chan Senior Member Posts: 633 Joined: Jul 2018
RE: (12C Platinum) Internal Precision Test
(12-31-2018 02:21 PM)Dieter Wrote:  ... The 12C Platinums works with 12 digits while the hardware can only display 10 of these.
In a way that's similar to, say, the TI-58/59: you see 10 digits but 13 are present.

So, HP 12C and 12C Platinum does totally different way of rounding ...
Is it still appropriate to compare internal precisions this way ?

Even if 12C had a million digits precision, due to rounding-back-to-10 digits, it still require 11 iterations.
12-31-2018, 04:24 PM
Post: #5
 Dieter Senior Member Posts: 2,398 Joined: Dec 2013
RE: (12C Platinum) Internal Precision Test
(12-31-2018 03:32 PM)Albert Chan Wrote:  So, HP 12C and 12C Platinum does totally different way of rounding ...
Is it still appropriate to compare internal precisions this way ?

Even if 12C had a million digits precision, due to rounding-back-to-10 digits, it still require 11 iterations.

?!?

The 12C is a 10-digit calculator with a 10-digit display.
The 12C Platinum is a 12-digit calculator with a 10-digit display.

The 12C Platinum does not round back to 10 digits.

Dieter
12-31-2018, 04:55 PM (This post was last modified: 12-31-2018 06:22 PM by Albert Chan.)
Post: #6
 Albert Chan Senior Member Posts: 633 Joined: Jul 2018
RE: (12C Platinum) Internal Precision Test
(12-31-2018 02:21 PM)Dieter Wrote:  Also let's be accurate about the term "internal precision". This usually referns to the precision of the calculator's internal (sic!) calculations. On HP calculators these are typically three more digits than what is exposed to the user, i.e. 13 dgits for 10-digit calculators or 15 digits for 12-digit devices.

I probably mis-understood Gamo's meaning of the word: internal precision.
This test really test the "exposed" digits. (10 for 12C, 12 for 12C Platinum)

More (non-exposed) internal precisions had almost no effect on iteration count.
From HP brochure, 12C Platinum have internal precisions of 15 digits.

Sorry for the noise ...
01-01-2019, 03:44 AM (This post was last modified: 01-01-2019 03:45 AM by Gamo.)
Post: #7
 Gamo Senior Member Posts: 497 Joined: Dec 2016
RE: (12C Platinum) Internal Precision Test
The Internal Precision as shown in brochure stated
is 15 digits.
The reason that 12C Platinum took more iterations because of the 15 digits internal precision which mean that this takes more iterations than the standard 12C and what shown on display is 10 digits.
This program just to proof of the numbers of iterations different between standard 12C and 12C Platinum not the accuracy of e .

Gamo
01-02-2019, 07:33 PM
Post: #8
 Dieter Senior Member Posts: 2,398 Joined: Dec 2013
RE: (12C Platinum) Internal Precision Test
(01-01-2019 03:44 AM)Gamo Wrote:  The Internal Precision as shown in brochure stated
is 15 digits.
The reason that 12C Platinum took more iterations because of the 15 digits internal precision which mean that this takes more iterations than the standard 12C and what shown on display is 10 digits.

Let's look at this a bit more closely.

The reason for the different number of iterations is not the 15-digit vs. 13-digit internal precision of both calculators. The program calculates with 12 and 10 digit precision, and this is what causes the difference.

The e approximation changes as long as 1/n! stays ≥ 5E–12 or 5E–10, respectively. This happens up to n=14 and n=12, respectively. This is what causes the different number of iterations. The fact that you determined 14 and 11 (instead of 12) is cause by random roundoff errors.

Dieter
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