HP Forum Archive 19

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12C+ Gamma Message #1 Posted by Gerson W. Barbosa on 25 June 2009, 10:49 a.m. Gamma(x+1) (0 <= x <= 39.4) 01 STO 0 24 RCL 2 02 . 25 y^x 03 6 26 / 04 y^x 27 STO+ 3 05 1 28 1 06 9 29 STO+ 2 07 * 30 GTO 12 08 STO 1 31 RCL 3 09 0 32 2 10 STO 2 33 SQRTx 11 STO 3 34 LASTx 12 RCL 2 35 / 13 RCL 1 36 / 14 x<=y 37 2 15 GTO 31 38 ENTER 16 x<>y 39 LN 17 2 40 / 18 * 41 RCL 0 19 1 42 1 20 + 43 + 21 RCL 0 44 y^x 22 y^x 45 / 23 2 46 GTO 00 12.12 R/S => 648,976,950.1 after 3.5 seconds (HP-12C+) On the 15C, 12.12 x! => 648,976,950.9 This is based on the following approximation I've come up with: Gamma(x+1) ~ SUM(k=0,inf,(2*k+1)^x/2^k)/(sqrt2/2*(2/ln(2))^(x+1)) Obviously, this is not the proper way to compute the Gamma function. It's just an example of what can be done on the fast 12C+. Just for an idea, it would take about 7 minutes to compute Gamma(40) on a normal 12C using this method. Gerson. -------------- The approximation is being used reversely, that is, it is meant to approximate this integer sequence, not the Gamma function: an ~ n!/(2*sqrt2)*(2/ln(2))^(n+1) I found it when thinking about John Smitherman's math joke turned into a problem. The numbers 2, 6, 34, 294... (which showed as the problem was getting progressively more complex) ought to mean something... I couldn't find it at OEIS but I found 1, 3, 17, 147..., the halved sequence. Eventually I found out the original sequence is the row sums of Sierpinski-Pascal-MacMahon triangle, whatever this might be. Next time someone starts a joke I'll try to just laugh, as everybody else :-) I just intended to put a computationally intensive programming problem to the 12C+ to see how it performed. For computing Gamma function on the 12C+ or the normal 12C and 12c Platinum there isn't anything better than Egan Ford's program: http://www.hpmuseum.org/cgi-sys/cgiwrap/hpmuseum/archv016.cgi?read=100275 See message #6. Edited: 26 June 2009, 10:30 a.m.

Re: 12C+ Gamma Message #2 Posted by Bart (UK) on 29 June 2009, 9:10 a.m., in response to message #1 by Gerson W. Barbosa Just for fun I thought I'd try it on the new 35s to see how it did (considering it's slighly(?) old CPU). 12.12 XEQ is about 30.5s with an answer of 648,976,950.899 (FIX = ALL). 40 XEQ is about 1 min. Bart

Re: 12C+ Gamma Message #3 Posted by Gerson W. Barbosa on 29 June 2009, 10:57 a.m., in response to message #2 by Bart (UK) 13 and 23 seconds, respectively, on the 33s which used to be my fastest RPN calculator. I wonder how nice it would be if they released a new 15C+. If it were as fast as the new 12C+, 60.5 CHS x! would take about 200 ms to display -1.527756e-97 instead of current 13 seconds. Gerson.

Re: 12C+ Gamma Message #4 Posted by Bart (UK) on 30 June 2009, 11:00 a.m., in response to message #3 by Gerson W. Barbosa A fast 15C would be a dream come true. I tried a HP-20S implementation: 01 LBL A 23 2 45 1/x 02 STO 0 24 + 46 * 03 y^x 25 1 47 2 04 . 26 = 48 = 05 6 27 y^x 49 y^x 06 = 28 RCL 0 50 ( 07 * 29 = 51 RCL 0 08 1 30 2 52 + 09 9 31 y^x 53 1 10 = 32 RCL 2 54 ) 11 STO 1 33 = 55 = 12 0 34 1/x 56 * 13 STO 2 35 * 57 2 14 STO 3 36 LAST 58 SQRT 15 LBL B 37 = 59 / 16 RCL 2 38 STO+ 3 60 2 17 INPUT 39 1 61 = 18 RCL 1 40 STO+ 2 62 1/x 19 x<=y? 41 GTO B 63 * 20 GTO C 42 LBL C 64 RCL 3 21 RCL 2 43 2 65 = 22 * 44 LN 66 RTN Compared to the original post, one can see the advantage of RPN here. Although this is probably not the most optimum. 12.12 XEQ A, 15.5s; 648,976,950.9 40 XEQ A, 30s Considering the program is 50% longer, that makes the 20S 3x faster than the 35s. Possibly the advantage of simple data types? Bart edited to try and make listing more readable Edited: 30 June 2009, 11:09 a.m.

Re: 12C+ Gamma Message #5 Posted by Raymund Heuvel on 29 June 2009, 6:35 p.m., in response to message #1 by Gerson W. Barbosa HP-71B with Math Pack a=time @ GAMMA(13.12); @ time-a results in 648976950.903 .23 thus 0.23 seconds .... The good old HP-71B ... BR Raymund Edited: 29 June 2009, 6:37 p.m.

Re: 12C+ Gamma Message #6 Posted by Gerson W. Barbosa on 29 June 2009, 7:50 p.m., in response to message #5 by Raymund Heuvel Quote: 648976950.903 .23 thus 0.23 seconds .... Haven't you missed a "1" before ".23"? 1.22 seconds on mine... ...and 2.04 seconds for GAMMA(-68.5). Gerson.

Re: 12C+ Gamma Message #7 Posted by Raymund Heuvel on 29 June 2009, 10:18 p.m., in response to message #6 by Gerson W. Barbosa a=time @ GAMMA(13.12); @ time-a execution time between 0.22 up to 0.26 a=time @ GAMMA(-68.5); @ time-a result = -1.527757e-97 execution time 1.01 (10 runs) FYI VER$ => HP71:1BBBB HPIL:1A MATH:1A FTH:1A EDT:A KBD:B I've tested it on three 71-B's with different memory sizes (16K up to 144K Ram), all about the same. Maybe you have a Plug-In or Lex File causing a lot of Interrupts or Polls? BR Ray

Re: 12C+ Gamma Message #8 Posted by Marcus von Cube, Germany on 30 June 2009, 4:42 a.m., in response to message #7 by Raymund Heuvel Mine must be on steroids: .09 seconds for GAMMA(13.12). It appears to be a very inaccurate measurement because several tries resulted in vastly disparate timing results, a few outcomes even with a negative sign.

Re: 12C+ Gamma Message #9 Posted by Gerson W. Barbosa on 30 June 2009, 7:51 a.m., in response to message #7 by Raymund Heuvel I was doing a=time @ GAMMA(13.12) @ time-a (without semicollon). Now I get 0.21 to 0.22 and 1.02 to 1.05 for GAMMA(-68.5). 1BBBB version also with a 32K RAM module and card reader. They have no effect on timing though. Thanks! Gerson.

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