Most of us are familiar with Mike Sebastian's standard 9 sin cos tan arctan arccos arcsin routine to compare the precision of assorted calculator makes. But it wasn't the first forensics sequence that I encountered -- many years ago (>40 at least!) I seem to recollect the following being used, but I can't recall which make or model's manual suggested it. The sequence is as follows (at least in RPN, degrees mode):

29 sin cos tan ln 1/x 1/x e^x arctan arccos arcsin.

It adds a few more operations, in other words.

I was wondering if anyone else remembers this and if so, where it might have come from?

(For the record, my WP34S returns a value of 29 in double precision mode, and my HP Prime gives 29.0000023889 -- exactly the same as my HP 27S -- and my TI 36X Pro yields 29.00000017.)

Here's a repeat of part of a post of mine from several years ago (which I can't find now to put in a link, and the search here does not work well) in response to a complaint that the HP-41 calculator did not have good accuracy because alog(log(60)) showed 59.999999950:

This is something slide-rule users will have a better understanding of. It's not really any bug in the machine, just a limitation of the number of digits it uses.

Consider: ln(60) is 4.09434456222, according to my HP-71B which uses more digits than the 41 does. Using only 10 digits, e^4.094344562 is 59.9999999867, and e^4.94344563 (ie, incrementing the least-significant digit by 1) is 60.0000000467, both quite a ways off from 60 in the 10th digit. IOW, there is no 10-digit number that will give you 60.00000000 when you take the natural antilog.

You find this more in for example cos(1°), only one digit, where you get .99984769516; but using only four digits of it to convert back to the angle, .9998 gets you over 14% high, at 1.1459..., and .9999 gets you .81029..., almost 19% low. There is no 4-digit number which, when taking its ACOS, will result in 1.0° (two digits), let alone 1.000° (four digits).

Commodore UK ran press advertisements using the 29 version of the forensics test, in October 1976. I remember seeing one, which I would have said was in the daily paper. But

this one is from New Scientist, a weekly:

[

attachment=9401]

[

attachment=9402]

But perhaps

from March 1976, in a US monthly "Popular Computing" we see the same test.

[

attachment=9403]

Here's an OCR:

Quote:THE ACCURACY TEST : Before you buy a scientific check the accuracy with this simple test : 29 sin cos tan VX in ex x2 tan - cos - sin - 1 : ? The 29 is degrees , of course , since no machine will invert the trig functions on 29 radians. The SR - 52 of Texas Instruments gives this result : 29 . 00001537 . Various Hewlett - Packard machines return values around 29 . 00xxxx , inasmuch as they carry their calculations to only 10 significant digits . The CBM ad fails to state just what the result is on their machine ...

which tells us that the idea again comes from a Commodore advertisement. Aha - but in fact,

in 1978, it is said to come from a UK advert!

Quote:In issue 46 we reported an advertisement by CBM Commodore , U . K . , Ltd . , which suggested that you test any " scientific " calculator by carrying out

(I see now that The Observer, a Sunday paper, and Private Eye, a fortnightly, also carried Commodore ads with this test.)

(04-19-2021 08:01 AM)EdS2 Wrote: [ -> ]Here's an OCR:

Quote:THE ACCURACY TEST : Before you buy a scientific check the accuracy with this simple test : 29 sin cos tan VX in ex x2 tan - cos - sin - 1 : ? The 29 is degrees , of course , since no machine will invert the trig functions on 29 radians. The SR - 52 of Texas Instruments gives this result : 29 . 00001537 . Various Hewlett - Packard machines return values around 29 . 00xxxx , inasmuch as they carry their calculations to only 10 significant digits . The CBM ad fails to state just what the result is on their machine ...

which tells us that the idea again comes from a Commodore advertisement. Aha - but in fact, in 1978, it is said to come from a UK advert!

The OCR is terrible! Here's a version you can figure out a little better:

Code:

`In Degree mode:`

29 sin cos tan √ ln e˟ x² tanˉ¹ cosˉ¹ sinˉ¹

(04-19-2021 06:08 AM)JimP Wrote: [ -> ]The sequence is as follows (at least in RPN, degrees mode):

29 sin cos tan ln 1/x 1/x e^x arctan arccos arcsin.

It adds a few more operations, in other words.

I was wondering if anyone else remembers this and if so, where it might have come from?

(For the record, my WP34S returns a value of 29 in double precision mode, and my HP Prime gives 29.0000023889 -- exactly the same as my HP 27S -- and my TI 36X Pro yields 29.00000017.)

The TI-36X Solar gives 28.99999928

In Degree mode, '29 sin cos tan √ ln e˟ x² tanˉ¹ cosˉ¹ sinˉ¹ the SR4190R the ad was about returns 29.00100887 the PR100 gets 29.085834

The Электроника МК-52 gives 28.934375 for the test.

(04-20-2021 07:12 AM)paul0207 Wrote: [ -> ]The Электроника МК-52 gives 28.934375 for the test.

Not so bad, my Novus Mathematician gives 30.22698

J-F

(04-19-2021 10:12 AM)toml_12953 Wrote: [ -> ]The OCR is terrible! Here's a version you can figure out a little better:

Code:

`In Degree mode:`

29 sin cos tan √ ln e˟ x² tanˉ¹ cosˉ¹ sinˉ¹

Thanks! It's unfortunate that we now have two different tests in this thread - one using reciprocal and the other using square root. So that makes it difficult to compare results.

(04-20-2021 07:48 AM)EdS2 Wrote: [ -> ]It's unfortunate that we now have two different tests in this thread - one using reciprocal and the other using square root. So that makes it difficult to compare results.

All these variants are based on the same quirk. they look complicate but the bottleneck is the COS ACOS sequence with an argument less than 1 degree (since it comes from SIN)

If I remove the COS and ACOS terms, even my Novus Mathematician gives 28.99983, which is really good for this very limited machine.

Just do .5 COS ACOS on various old machines and see what happens (modern machines will have no problem of course).

J-F

(04-20-2021 08:05 AM)J-F Garnier Wrote: [ -> ]All these variants are based on the same quirk. they look complicate but the bottleneck is the COS ACOS sequence with an argument less than 1 degree (since it comes from SIN)

Is this why

versin was invented ?

(04-19-2021 06:29 AM)Garth Wilson Wrote: [ -> ]There is no 4-digit number which, when taking its ACOS, will result in 1.0° (two digits), let alone 1.000° (four digits).

Using versin / arcversine, we do well with 4-signficant digits.

lua> versin = function(x) return 2*sin(x/2)^2 end

lua> arcversin = function(x) return 2*asin(sqrt(x/2)) end

lua> versin(rad(1)) -- 1 - cos(1°)

0.00015230484360876083

lua> deg(arcversin(0.0001523)) -- acos(1 - 0.0001523), then rad→deg

0.999984098436689

(04-19-2021 06:08 AM)JimP Wrote: [ -> ]Most of us are familiar with Mike Sebastian's standard 9 sin cos tan arctan arccos arcsin routine to compare the precision of assorted calculator makes. But it wasn't the first forensics sequence that I encountered -- many years ago (>40 at least!) I seem to recollect the following being used, but I can't recall which make or model's manual suggested it. The sequence is as follows (at least in RPN, degrees mode):

29 sin cos tan ln 1/x 1/x e^x arctan arccos arcsin.

It adds a few more operations, in other words.

I was wondering if anyone else remembers this and if so, where it might have come from?

(For the record, my WP34S returns a value of 29 in double precision mode, and my HP Prime gives 29.0000023889 -- exactly the same as my HP 27S -- and my TI 36X Pro yields 29.00000017.)

I tried this on my HP 50gs running newRPL firmware. Just for fun, I set the precision to 1000 digits.

The result is displayed as

29.0. with the trailing '.' indicating not exact). When I subtract 29, the residual is -5.8044.E-994 which seems well within the ballpark.

My HP 48G gives me 29.0000023889. This apparently is a standard HP result.

(04-20-2021 08:05 AM)J-F Garnier Wrote: [ -> ]Just do .5 COS ACOS on various old machines and see what happens (modern machines will have no problem of course).

Cosine of small angles is also difficult for slide rules, unless it is first converted to radians and then the following approximation is used:

Paul

(04-22-2021 04:28 AM)paul0207 Wrote: [ -> ]Cosine of small angles is also difficult for slide rules, unless it is first converted to radians and then the following approximation is used:

Paul

I think British Thornton tried to address this issue by including "differential trig" scales on some of their slide rules. They allowed you to compute the sine of angles very close to 90 (or cosine of angles close to 0) with a wee bit more accuracy than typical trig scales.

Another way to compare is to note that on a typical 10" slide rule, the physical distance between sin 80 and sin 90 is about 1/16th of an inch, while on the 10" British Thornton the distance between the same two values on its differential trig scale is about 7/16th of an inch, which provides quite a bit more room for divining values.

(04-24-2021 06:31 AM)Benjer Wrote: [ -> ]I think British Thornton tried to address this issue by including "differential trig" scales on some of their slide rules. They allowed you to compute the sine of angles very close to 90 (or cosine of angles close to 0) with a wee bit more accuracy than typical trig scales.

Another way to compare is to note that on a typical 10" slide rule, the physical distance between sin 80 and sin 90 is about 1/16th of an inch, while on the 10" British Thornton the distance between the same two values on its differential trig scale is about 7/16th of an inch, which provides quite a bit more room for divining values.

Thanks for the info.

I found this article by the Oughtred Society that mentions the "differential scales" used by British Thornton and the "evenly spaced scales" by Australian W&G slide rules:

https://osgalleries.org/journal/pdf_file...9.1P33.pdf