Calcuator forensics history question

04192021, 06:08 AM
Post: #1




Calcuator forensics history question
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.) 

04192021, 06:29 AM
Post: #2




RE: Calcuator forensics history question
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 HP41 calculator did not have good accuracy because alog(log(60)) showed 59.999999950:
Consider: ln(60) is 4.09434456222, according to my HP71B 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 leastsignificant digit by 1) is 60.0000000467, both quite a ways off from 60 in the 10th digit. IOW, there is no 10digit 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 4digit number which, when taking its ACOS, will result in 1.0° (two digits), let alone 1.000° (four digits). http://WilsonMinesCo.com (Lots of HP41 links at the bottom of the links page, http://wilsonminesco.com/links.html ) 

04192021, 08:01 AM
(This post was last modified: 04192021 08:30 AM by EdS2.)
Post: #3




RE: Calcuator forensics history question
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:
But perhaps from March 1976, in a US monthly "Popular Computing" we see the same test. 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.) 

04192021, 10:12 AM
(This post was last modified: 04192021 10:14 AM by toml_12953.)
Post: #4




RE: Calcuator forensics history question
(04192021 08:01 AM)EdS2 Wrote: Here's an OCR: The OCR is terrible! Here's a version you can figure out a little better: Code: In Degree mode: Tom L Cui bono? 

04192021, 10:16 AM
Post: #5




RE: Calcuator forensics history question
(04192021 06:08 AM)JimP Wrote: The sequence is as follows (at least in RPN, degrees mode): The TI36X Solar gives 28.99999928 Tom L Cui bono? 

04192021, 04:39 PM
Post: #6




RE: Calcuator forensics history question
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


04202021, 07:12 AM
Post: #7




RE: Calcuator forensics history question
The Электроника МК52 gives 28.934375 for the test.


04202021, 07:37 AM
Post: #8




RE: Calcuator forensics history question  
04202021, 07:48 AM
Post: #9




RE: Calcuator forensics history question
(04192021 10:12 AM)toml_12953 Wrote: The OCR is terrible! Here's a version you can figure out a little better: 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. 

04202021, 08:05 AM
Post: #10




RE: Calcuator forensics history question
(04202021 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). JF 

04202021, 02:26 PM
Post: #11




RE: Calcuator forensics history question
(04202021 08:05 AM)JF 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 ? (04192021 06:29 AM)Garth Wilson Wrote: There is no 4digit 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 4signficant 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 

04222021, 12:23 AM
Post: #12




RE: Calcuator forensics history question  
04222021, 02:31 AM
Post: #13




RE: Calcuator forensics history question
(04192021 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): 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.E994 which seems well within the ballpark. My HP 48G gives me 29.0000023889. This apparently is a standard HP result. Central PA, USA 16C, 48G, 39gs(newRPL), 40gs(newRPL), 50g(newRPL), Prime G2 

04222021, 04:28 AM
(This post was last modified: 04222021 04:59 AM by paul0207.)
Post: #14




RE: Calcuator forensics history question
(04202021 08:05 AM)JF 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 

04242021, 06:31 AM
Post: #15




RE: Calcuator forensics history question
(04222021 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: 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. 

04252021, 01:59 AM
Post: #16




RE: Calcuator forensics history question
(04242021 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. 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 

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