Welcome back, Palmer!

Did you happen to notice my post of 24 December, which revisited one of your archived posts?

http://www.hpmuseum.org/cgi-sys/cgiwrap/hpmuseum/forum.cgi?read=130050#130050

-- KS

Karl:

I am in the process of catching up since I was distracted (actually, a lot more than distracted) by my wife's major surgery.

I have taken a quick look at the commentary on the TI-55II that you mentioned. It will take me some time to digest it thoroughly. However, I should note that you stated that

Quote:

The TI-55II was an 8-digit model.

My recollection is that the TI-55II displays eight digits but carries eleven digits internally. That would make sense with the problem described in your commentary where the mean is displayed as 6 but subtracting six from the displayed value yields -1 E-10.

Woerner Joerg's site lists the internal capability of the TI-55II as eleven digits.

Palmer

Hi again, Palmer --

Happy holidays, and we all hope that your wife's surgery and recovery turns out for the best.

Quote:

My recollection is that the TI-55II displays eight digits but carries eleven digits internally.

Yes, I'm sure that you're correct, and those three extra guard digits contain the small inaccuracy from recalculation of the mean. However, it's possible for those calculations to be exact if the data are entered in ascending or descending order: {4, 5, 6, 7, 8} or {8, 7, 6, 5, 4}. The only intermediate mean average of up to five input data that is __not__ exactly representable using two or fewer decimal points, is the mean of three data having an odd sum.

On a TI-55II or similar, try entering {4, 6, 7, 8, 5} using stat summation, or manually calculate

(((4 + 6 + 7)/3)*3 + 8 + 5)/5 - 6.

I get -1E-09 on my LED TI-30, which also has an eight-digit mantissa and three guard digits.

Reducing the possibility of overflow with an eight-digit mantissa was probably the reason for the TI-55II's statistical-summation methods. The guard digits were not to be considered reliable; on the LED TI-30, they certainly *weren't*:

http://www.hpmuseum.org/cgi-sys/cgiwrap/hpmuseum/archv016.cgi?read=107358#107358

-- KS

*Edited: 31 Dec 2007, 2:33 a.m. *

The NewEgg newsletter, which you have to subscribe to, has a special until Jan 2. $51.99 with free shipping and no tax to some states (like mine). I was hoping to get one for Christmas but, alas, nothing. First time in 30 years I was disappointed to not get a Christmas present I really wanted.

Karl:

The surgery was on the Friday before Christmas and my wife was home by Christmas morning. Now, there is a REAL Christmas present!

Quote:

Yes, I'm sure that you're correct, and those three extra guard digits contain the small inaccuracy from recalculation of the mean. However, it's possible for those calculations to be exact if the data are entered in ascending or descending order: {4, 5, 6, 7, 8} or {8, 7, 6, 5, 4}. The only intermediate mean average of up to five input data that isnotexactly representable using two or fewer decimal points, is the mean of three data having an odd sum.

The ascending sequence is exactly the one that I described on V8N1P26 od TI PPC Notes and which yields an error of -1E-10. For the descending sequence the error is +1E-10.

Either sequence yields exactly 6 on the TI-55 which was the machine that the TI-55II was supposed to replace. Both the TI-55 and the TI55II use 11 digits internally and display 8. Neither one has the non-commutative multiply problem seen with the TI-57, TI-58 and TI59. However, there is something different going on in the mathematics of the two machines. For 2 divided by 3 the TI-55 yields an internal value of 0.66666666667 while the TI-55II yields a value of .66666666666. The TI-55 yields similar rounding of the least significant internal digit for other simple tests while the T-55II yields truncated results. I will try to dig out some more tests of rounding and report the results.

I was surprised by the internal rounding in the T-55 since I had thought that the TI-95 was the first TI machine with such rounding.

Are you aware of any specific test results of the accuracy of the multiplication algorithm of the TI-30, TI-55, TI-55II, etc.?

Palmer

Palmer --

Yes, good news indeed!

Unfortunately, I have no insights to offer regarding accuracy and bugs of the legacy TI models, other than what I have already detailed with the original LED TI-30 -- the first calc I owned in 1977, and for which I now have an identical replacement.

That division result of the TI-55II may be a symptom of the inaccuracies, though.

-- KS

Karl:

You wrote:

Quote:

That division result of the TI-55II may be a symptom of the inaccuracies, though.

Some linited testing shows that TI-55 also provides rounding to the eleventh digit of the mantissa for multiplication. The TI-55II does not round to the eleventh digit for multiplication but rather truncates. The TI-30 truncates some of the time, but sometimes does something else. For example, for th product 2.222223 x 3.333333:

Exact 13 digits 7.407409259259TI-55 11 digits 7.4074092593

TI-55II 11 digits 7.4074092592

TI-30 11 digits 7.4074092591

If one emulates the statistics routine which finds the mean with the TI-55II in user memory of the TI-55 with the program

R/S - RCL 1 = EXC 0 + 1 = EXC 0 x RCL 0 1/x + RCL 1 = STO 1 RST

then after clearing memories 0 and 1 and pressing reset one can calculate by entering the values and pressing R/S. The machine will stop with the current mean in the display and ready for the next input. For the 4, 5, 6, 7, 8 sequence or it's reverse the mean will be exactly six.

One can emulate the statistics routine in the user memory of the TI-55II with the same routine but one does not need the leading R/S because with the TI-55II the RST command returns the machine to program origin and stops.. For the 4, 5, 6, 7, 8 sequence the mean will not be exactly six but will be in error by 1E-10.

The change the response to a RST command caused unhappiness in the TI user community because it eliminated the capability to run iterative loops. That's a much worse "improvement" than changing the size or location of the ENTER key. Of course, that wasn't the major complaint since the TI-55II and it's companion machines such as the BA-55 and TI-57LCD has the world's worst key bounce.

You can get free shipping from Amazon too.

It will be slower, of course.

Some may prefer the service from Newegg, but that's another subject.

Hi again, Palmer --

Quote:

The TI-30 truncates some of the time, but sometimes does something else. For example, for the product 2.222223 x 3.333333:

Exact 13 digits 7.407409259259

TI-55 11 digits 7.4074092593

TI-55II 11 digits 7.4074092592

TI-30 11 digits 7.4074092591

Yup, there were some math problems with the legacy TI-30 and the TI-55II.

A summation "trick" for calculating very small quadratic-equation discriminants with improved accuracy is given in the HP-15C Advanced Functions Handbook. It allows A - BC to be calculated to 13-digit extended precision. You had ported this method to other calculators, as described in one or more contributions to the HP Articles Forum.

Here are -- bonus! -- __three__ ways to obtain the three lowest-order decimal digits of the the exact 13-digit product of complete operands using the 10-digit HP-15C, by exploiting its extended-precision arithmetic to see "7409259__259__".

Statistical Summation Matrix Residual Matrix Multiplication

--------------------- --------------- ---------------------CLEAR SIGMA 1 1

7.40 ENTER ENTER

STO 7 DIM A 2

2.222223 DIM B DIM A

ENTER DIM C x<>y

3.333333 MATRIX 1 DIM B

SIGMA- 2.222223 MATRIX 1

RCL 7 STO A USER

CLEAR PREFIX 7.40 2.222223

STO B STO A

3.333333 -1

STO C STO A

RCL MATRIX A 3.333333

RCL MATRIX C STO B

RESULT B 7.40

MATRIX 6 STO B

RCL B RCL MATRIX A

CLEAR PREFIX RCL MATRIX B

RESULT C

*

RCL C

CLEAR PREFIX

The statistical-summation method will also work on the predecessor HP-34C and HP-11C, with register R5 instead of register R7, and using "MANT" instead of "CLEAR PREFIX" on the HP-34C.

-- KS

*Edited: 6 Jan 2008, 12:58 a.m. after one or more responses were posted*

Quote:You mean quadratic-equationA summation "trick" for calculating very small quadratic-equation determinants...

*discriminants*, don't you?

A somewhat more direct way to get an extra 3 digits on the HP48 and its relatives is:

[ 2.222223 -1. ] [ 3.333333 7.0474 ] DOT

But the 15-form rounding isn't always round to even. It's different for 15-form multiplication than for 15-form addition, so the extra digits may not be properly rounded to even.

I wonder what the HP15 does with the extended products and sums?

Quote:

You mean quadratic-equationdiscriminants, don't you?

Oops! Yes, I did. My post is corrected.

Quote:

A somewhat more direct way to get an extra 3 digits on the HP48 and its relatives is:[ 2.222223 -1. ] [ 3.333333 7.0474 ] DOT

You mean, 7.__40__74, don't you? *(Gotcha!)*

Quote:

I wonder what the HP15 does with the extended products and sums?

Uses each of them for calculating the final result, which is rounded to 10 significant digits for storage in the stack register or numbered register (unless I'm not understanding the question).

-- KS

*Edited: 6 Jan 2008, 12:29 a.m. after one or more responses were posted*

Quote:Rounded how, was what I was asking. Rounded to even, or just truncated?

Uses each of them for calculating the final result, which is rounded to 10 significant digits for storage in the stack register or numbered register (unless I'm not understanding the question).-- KS

There was a thread a few years ago on comp.sys.hp48 about this. The 15-form arithmetic on the later Saturn machines isn't rounded to even. It wasn't rounded to even on the HP71 either, but for some reason, they did it differently on the HP48 than they did on the HP71.

Quote:

Rounded how, was what I was asking. Rounded to even, or just truncated?

It seems that you're wondering whether the 13-digit extended-precision results are simply truncated (i.e., all digits correct in their places), or are rounded with or without bias (e.g, "to even, to odd, always up or always down").

In order for for the "correct" rounding to be always known, it would seem that additional digits beyond the 13th must be computed. I'm not sure if that is even done. However, I did check out the rounding of a 13-digit internal result to the final 10-digit result. The HP-15C appears to round the 10th significant digit up to the digit of greater magnitude. So, the issue of any bias in the 13th digit would seem of minimal importance.

FIX 9

9.000000001

5E-10

+ 9.000000002

LSTx

+ 9.000000003

LSTx

+ 9.000000004

CHS

LSTx

- -9.000000005

LSTx

- -9.000000006

LSTx

+ -9.000000006

etc...

-- KS

*Edited: 6 Jan 2008, 7:41 p.m. *