Just a curious mathematical result

03052018, 10:54 PM
Post: #1




Just a curious mathematical result
Hi all,
Just to let you know about a curious mathematical result I discovered recently. In radians: Sin(9*(Sin(1) + Cos(40))) = 0.999999999999999830826985368... which differs from the integer 1 by about 1e16. If you compute it on a 12digit machine (such as the HP71B) y will probable get evaluated as exactly 1. The correct value requires more than 12 digits, in a pinch the Windows calculator will do. ¿ Nice, isn't it ? V. . All My Articles & other Materials here: Valentin Albillo's HP Collection 

03052018, 11:33 PM
Post: #2




RE: Just a curious mathematical result
This little problem makes me appreciate Longfloat and newRPL very much.


03052018, 11:33 PM
(This post was last modified: 03062018 12:12 AM by Valentin Albillo.)
Post: #3




RE: Just a curious mathematical result
(03052018 11:08 PM)Mike (Stgt) Wrote: Should be added here. (sorry for the typos there in my original post, virtual keyboards can be a chore and I didn't have time to properly proofread) Yes, I think you're right so I've just contacted the people in charge of the web page you mentioned to let them know and perhaps consider including it there. By the way, Mike, did you have a look at the PDF document I suggested in my reply to your post on Namir's thread, namely: AMS  Computing square root of 2 to 1,000,000 digits The document discusses algorithms used in the past to compute square roots to thousands of digits and in particular the one used to reach 1,000,000 digits. Also, to check your very own implementation simply google "square root of 2 to 1 million digits" and you'll immediately find a link to the full result. Regards. V. . All My Articles & other Materials here: Valentin Albillo's HP Collection 

03062018, 07:07 AM
Post: #4




RE: Just a curious mathematical result
A question would be. How did you get that formula in the first place? It is not a common one.
Wikis are great, Contribute :) 

03062018, 01:59 PM
Post: #5




RE: Just a curious mathematical result
Works on SM DM42 but gives "1" in Prime "home" rpn and "1." in Prime CAS
Esben 28s, 35s, 49G+, 50G, Elektronika MK52 & MK61 

03062018, 02:10 PM
Post: #6




RE: Just a curious mathematical result
(03062018 01:59 PM)DA74254 Wrote: Works on SM DM42 but gives "1" in Prime "home" rpn and "1." in Prime CAS Well, it's the correct answer given the HP Prime's number of digits / bits in the two modes. Software Failure: Guru Meditation  Antonio IU2KIY 

03062018, 03:02 PM
Post: #7




RE: Just a curious mathematical result
...and the WP34S in doubleprecision mode yields
0.9999999999999998308269853686957601 Jake 

03062018, 03:05 PM
Post: #8




RE: Just a curious mathematical result
(03062018 02:10 PM)TheKaneB Wrote:(03062018 01:59 PM)DA74254 Wrote: Works on SM DM42 but gives "1" in Prime "home" rpn and "1." in Prime CAS Yes, I'm aware of that. But if you calculate a large number, say; 99^20, you may hit "view" and you can obtain a number 100's of digits long. Then 1/x and you get a resiprocal of that large number. Though, the "1." in CAS is absolute. You won't be able to evaluate it, only obtain the approx version of the formula stated by OP. Esben 28s, 35s, 49G+, 50G, Elektronika MK52 & MK61 

03062018, 04:15 PM
(This post was last modified: 03062018 04:32 PM by jebem.)
Post: #9




RE: Just a curious mathematical result
(03062018 01:59 PM)DA74254 Wrote: Works on SM DM42 but gives "1" in Prime "home" rpn and "1." in Prime CAS Yes, HP Prime on CAS mode gives me 1. for approx(SIN(9*(SIN(1)+COS(40)))) However doing 1Ans gives me 3.5527136788 E15 So, not exactly the same answer of absolute 1 as got on Home mode, where 1Ans gives exactly 0. Further quick searching on google shows a few answers for the above CAS result of 3.5527136788 E15 for instance here. Apparently this value corresponds to 8 ^ 16 or 16 ^ 12 or 64 ^ 8 (i didn't take the time to check it). Tim and Cyrille have been posting some information on the numbers internal representation being different on Home and on CAS modes, and that can explain the different results when trying to get the hidden guard digits on each mode. Jose Mesquita RadioMuseum.org member 

03062018, 04:35 PM
Post: #10




RE: Just a curious mathematical result
(03062018 03:05 PM)DA74254 Wrote:(03062018 02:10 PM)TheKaneB Wrote: Well, it's the correct answer given the HP Prime's number of digits / bits in the two modes. Hi, The Prime can do large integers, but for floating point numbers will always be 12 digit accuracy (actually CAS is double precision floating point, which could be 15 digits  but i'm not sure the developers will guarantee this). . 

03062018, 04:44 PM
(This post was last modified: 03062018 04:51 PM by BartDB.)
Post: #11




RE: Just a curious mathematical result
(03062018 04:15 PM)jebem Wrote:(03062018 01:59 PM)DA74254 Wrote: Works on SM DM42 but gives "1" in Prime "home" rpn and "1." in Prime CAS 1  sin(9*sin(1)+cos(40))) should yield 1.69173...E16, so the Prime is quite wrong. CAS approx. is double precision floating point which gives 15 digit accuracy (17 at most, but for Prime CAS I have found it to be seldom better than 15). However, as you have demonstrated, more digits can be found by subtracting the more significant digits  but I have found they are usually wrong. So I would not rely on it. . 

03062018, 04:49 PM
(This post was last modified: 03062018 04:54 PM by BartDB.)
Post: #12




RE: Just a curious mathematical result
(03052018 11:08 PM)Mike (Stgt) Wrote:(03052018 10:54 PM)Valentin Albillo Wrote: Hi all, These type of equations should be called "Rube Goldberg methods of obtaining almost integers" . 

03062018, 05:53 PM
Post: #13




RE: Just a curious mathematical result
Or weird approximations of pi.
18*(sin(1) + cos(40)) ~= pi 

03072018, 11:42 AM
Post: #14




RE: Just a curious mathematical result  
03072018, 02:23 PM
Post: #15




RE: Just a curious mathematical result  
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