A friend pointed me at this article about 355/113 and it's remarkable closeness to pi. I thought some of you might find it interesting.
Dave
[OT] 355/113 and pi

07252012, 04:53 PM
A friend pointed me at this article about 355/113 and it's remarkable closeness to pi. I thought some of you might find it interesting. Dave
07252012, 05:06 PM
That was a very interesting read! Instead of memorizing a fraction that is "close enough" to pi, I just went and memorized 3.1415926535897932384626433 and yes that is all from memory... :)
07252012, 05:55 PM
There is an answer to this issue (that that article doesn't have). The continued fraction expansion for pi is 3; 7, 15, 1, 292... which works out to 3, 22/7, 333/106, 355/113, 103993/33102,... There's no mystery where 355/113 comes from; it's one of the convergents of the continued fraction. http://en.wikipedia.org/wiki/Convergent_(continued_fraction)
07252012, 06:47 PM
Excellent catch. In case you don't wanna bother with wiki, I copied the pertinent info here Every real number can be expressed as a regular continued fraction in canonical form. Each convergent of that continued fraction is in a sense the best possible rational approximation to that real number, for a given number of digits. Such a convergent is usually about as accurate as a finite decimal expansion having as many digits as the total number of digits in the nth numerator and nth denominator. For example, the third convergent 333/106 for π (Pi) is roughly 3.1415094, which is not quite as accurate as the 6digit 3.14159; the fourth convergent 355/113 = 3.14159292 is more accurate than the 6digit decimal.
Edited: 25 July 2012, 6:49 p.m.
07252012, 08:07 PM
I wrote some FORTRAN programs for an IBM 360 mainframe in the 1970s that used 355/113 for pi, since it was accurate to within the machine accuracy. Some of these programs are still in use today on PCs, and 355/113 is a sufficiently accurate representation of pi for the purposes of the computations.
07252012, 09:31 PM
Yep, I never needed more accuracy either actually. I learned to code in FORTRAN on a DEC VAX 11/780 in the early eighties myself.
07252012, 11:54 PM
There was a post here a while ago (thus, it's in the archives somewhere) about pi to hundreds of thousands or millions of decimal places. Within the thread, someone mentioned a website which contains a searchable and browsable representation of pi. The image is a colourcoded representation, with each digit represented by a different coloured pixel. You could also move your mouse around this picture and the hypertext would display the digits of pi where you were browsing.
07262012, 12:13 AM
Quote:...83279 (from memory as well  which shows you what happens to bored teenagers in high school English classes. At least this one. Couldn't diagram a sentence to save my life, though.) But back OT: as you noted below the approximation 355/113 gives more than six digits accuracy and is arguably easier to remember:  First three odd integers, doubled: 113355  Split in the middle and divide One of my math professors had worked for NASA and brought this topic up in a numerical analysis class. He said they were computing trajectories for the Apollo 8 flight using two programs but kept getting different results. The problem turned out to be that "some &#$! engineer" had put 22/7 in for pi in one of them. "At 240,000 miles it makes a difference." I've always remembered that!
Edited: 26 July 2012, 12:20 a.m.
07262012, 01:45 AM
pi and other rationalnumber approximations I've found for use in scaledinteger math: http://wilsonminesco.com/16bitMathTables/RationalApprox.html
07262012, 02:55 AM
Quote: I teach at a K12 school. A couple of months ago at the elementary school talent show, a kid got up and recited pi to 50 digits. It was both impressive and entertaining. wes
07262012, 03:47 AM
In 1976 I found this while toying around with my beloved HP25:
07262012, 07:36 AM
Interesting... The BaileyBorweinPlouffe (BBP) formula is also interesting :
On the 50G
« 0 DUP ROT FOR k 5 ~PI '47/15+53/6552+829/5026560+79/15590400+857/4561108992+1901/244729774080' >NUM 3.14159265323 I computed 2000 digits of PI with this formula on my 50G
The new HP39gII give for PI with the embedded a b/c function : Same result with >Q on the 50G wich convert any real to a close rationnal number
Edited: 26 July 2012, 7:51 a.m.
07262012, 12:12 PM
I've read the BBP formula is used to compute individual hexadecimal digits of pi without calculation all the previous ones. There is a Mathcad implementation of the algorithm here, but I have no idea on how to port it to RPL.
Quote: How long did that take? Not too long, I presume. One year ago I computed pi to one thousand places on the HP 50g in slightly less than three hours, using a modified Archimedes algorithm and a thirdparty extended precision libray. That's a lot of time, but the original method would have required five times as much! http://www.hpmuseum.org/cgisys/cgiwrap/hpmuseum/archv020.cgi?read=188443 Regards, Gerson.
07262012, 02:35 PM
That's pretty close to 6317674554 / 2010978267 = 3.14159265551....
And a resounding "NO! We're not strange...the rest of the world is!" Edited: 26 July 2012, 2:36 p.m.
07262012, 03:40 PM
How did you get the extra 1900+ digits when the 50G only calculates to less than 16 digits of precision?
07262012, 04:34 PM
Quote:My 50g pies: http://sense.net/~egan/hpgcc/#Example:%20%20π%20Shootout Fastest uses FFTs. 8109 digits in 36 seconds. Of course I used C.
07262012, 04:35 PM
50G is only limited by memory for integers.
07262012, 04:52 PM
Hi Gerson, here is the program for 2000 decimals.
« It's only 207 Bytes. It requires no extension precision library. It only use the 'infinite' integer possibility of the 50G. In this case, it works with a 2000 digits integer number. The idea is : 10^2000 * * 10^2000 I evaluate the Plouffe formula with the 50G and transform it in full RPN (but it is not obvious it's faster that the algebraic, I dont try, but there is sometimes surprises with this !) Must run in exact mode, uncheck LASTARG With full speed emu48, it takes 18 minutes. Much more on a real 50G (many hours ?) (the emulator is much faster that the real calc with long integer, more than for other calulations...)
The number of iterations (here 1701 for 2000 décimals) is calculated with : For example, if you want 200 decimals to test, just change :
2000 ALOG > 200 ALOG So it possible to modify the program to ask the number or decimal you want... I tried with 5000 but it seems that a 5000 digits integer is too big for the 50G (Just try 5000 ALOG, you will get an error) ! It is very 'economic' in memory because 1 digit is only a nibble ( 1 octet for 2 digits !) PS : I deleted some library on my EMU48 and try again with 5000 digits.... My computer will work this night ;) PS : Gerson in your link, there is a PASCAL version. Seems very easy to use it for the 39Gii (copy and paste and few change !) What does the :nn here WriteLn(n:4,a:22:17,b:22:17); ?
Edited: 26 July 2012, 5:22 p.m.
07262012, 05:26 PM
Thanks for this ! It's very very interesting. Not only for PI but for yours explanations about "C" and HP50G !
I have the idea than (in theory !) it's possible to calculte more digits in RPL than in C but it will take more more time (probably too much to be really usable)!! My 5000 RPL PI digits is running ... Wait and see ;) Edited: 26 July 2012, 5:28 p.m.
07262012, 06:32 PM
Quote:Not with a single set of batteries. :) Connect up your USB port. If you read my page you'll see that I did 200,000 digits in ~17.5 hrs. In C @ 75MHz.
07262012, 06:54 PM
Thanks for your explanations!
Quote: That's just number formatting: n:4 > 4 printing positions for integer variable n a:22:17 > 22 positions for real variable a, 17 of which reserved for the fractional part.  P.S.: Here is a C++ version, for clarity:
#include <stdio.h>
Edited: 26 July 2012, 9:12 p.m.
07262012, 07:05 PM
Quote: That's really very impressive! Of course I wasn't trying to break any record with a 2,500 year old algorithm :) Also very interesting is your Pi Day Rematch: Apple II vs. HP41C last year. Were I brave enough, I would make my 8bit MSX computer join the fight :)
07262012, 07:55 PM
Quote: I would rephrase this as "Pi fans do strange things, don't they?" :) The other day I presented the following 12digit approximation involving ln(2) and e:
Jim Markovitch, whom I also showed it to, not only improved it to 16 decimal places but rewrote it in a very interesting way:
07262012, 08:59 PM
Still not clear. Are you saying that the 1900+ extra digits were calculated by one of the above algorithms one at a time?
07272012, 02:32 AM
Yes I saw that : really impressive on the calc at 75MHz! I've got my 5000 digits this night (but with emu48). Less than 8 hours but I don't know exactly (I runned with EVAL instead of TEVAL) I've an idea for a much smaller (!) and perhaps quicker program (User RPL) Gilles
Edited: 27 July 2012, 2:37 a.m.
07272012, 04:23 AM
Lighter and faster USR RPL :
Here for 2000 digits : « 181 Bytes and 371 sec with EMU48 for 2000 digits on my old laptop It use Plouffe formula '(1/(16^k))*((((4/((8*k)+1))(2/((8*k)+4)))(1/((8*k)+5)))(1/((8*k)+6)))' EVAL EXPAND > '(k^2*120+151*k+47)/((512*k^4+1024*k^3+712*k^2+194*k+15)*2^(4*k))' EDIT 1 : I just realise that 1701 is less than 2+log(8n)+n*log(16). I found this formula on the web after some tests for the value of the iterations. With 1701 the 4 or 5 lasts digits are false. EDIT 2 : 5000 digit in 7500 sec (~ 2 hours) with EMU48. The last four digits are wrong again curious
EDIT 3 : 10.000 in progress ;) But 200.000 will takes years or centuries ! :O :D IQUOT use lot of time with so big numbers !
Edited: 27 July 2012, 7:29 a.m.
07272012, 04:56 AM
Hi Matt, see my previous post and new algorithm. The idea is (for example ) : 1/3 ~= 0.3333333333 but we want to work in integer because the 50G can work with very big numbers in integer (in fact I dont know the limite....) Suppose we want 5 decimals for 1/3 but we want to work in integer only : 100000*(1/3) = 100000/3 ~= 100000*0.333333 > 33333(.333) With the 50G : (100000 * 1) IQUOT 3 @ IQUOT = Integer quotient on the 50G > 33333 (the 5 first decimals ) << 1000 ALOG 3 IQUOT >> > 3333...3333 (one hundred'3') And the HP50 can handle numbers like 5000 ALOG (1 followed by 5000 zero , 10^5000, which is much more thans the number a atoms in the Universe 10^80 (as far we know !) PS :I cannot test now, but '\GS(k=0,50,IQUOT(ALOG(64)*PEVAL([ 120 151 47 ],k), PEVAL([ 512 1024 712 194 15 ],k)*16^k)))' could give then 64 first decimal of PI. Not sure it will work
Edited: 27 July 2012, 5:37 a.m.
07272012, 10:01 AM
Quote:Yes. Arbitrary precision math often uses large integers or better stated, an array of integers.
07272012, 10:42 AM
Hi here is a 50G UsrRPL program to calculate the Nth digits of PI in hexadecimal. It is bases on the french explanation of wikipedia
« > n a « 0 n `n+15` FOR k `16^(nk)/(8*k+a)` + NEXT » >NUM » 'B' STO Just type n PIn to get the hexa digits for n1 to n+3 For example 200 PIn
Edited: 28 July 2012, 6:13 p.m. 
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