HP Forum Archive 19

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sum of the first n integers Message #1 Posted by Gerson W. Barbosa on 30 May 2010, 5:28 p.m. Most everybody here knows the story of young Gauss being asked by his teacher to add together the first 100 natural numbers and how he did it quickly in a very genial way. Lacking his genius, I will do it the old-fashioned way, summing the numbers one by one, not only the first 100 of them, but as much as I can do in 60 seconds using some different programmable calculators: HP-12C HP-33s HP-34c 01 1 G0001 LBL G 001 LBL A 02 ENTER G0002 1 002 1 03 ENTER G0003 ENTER 003 ENTER 04 1 H0001 LBL H 004 LBL 1 05 + H0002 1 005 1 06 + H0003 + 006 + 07 STO 0 H0004 + 007 + 08 LST x H0005 STO A 008 STO 0 09 GTO 04 H0006 LAST x 009 LST x H0007 GTO H 010 GTO 1 Execute the program and start your chronometer. When exactly one minute has elapsed, press the R/S key, recall the sum S from the memory register and compute n = (sqrt(8*S + 1) - 1)/2 or INT(e^((ln(S) + ln(8))/2 - ln(2))), in case 8*S might exceed the number of digits in the display. The following results were obtained: calculator S n HP-12c+ 58 001 835 10 770 HP-33s 7 610 851 3 901 HP-12c 14 365 169 HP-34c 6 441 113 These can be used to be benchmark the performance of the calculators. For instance, I gather from the table my HP-12c+ is 63.7 times faster then the ordinary 12c (10770/169) and my 33s is 34.5 times faster than my 34c. The latter agrees with other benchmarks: Tony Thimet's Benchmark HP-33s: 44.0 HP-34c: 1.3 -> 33.85x Xerxes's Benchmark HP-34c: 4390 s HP-33s: 131 s (my units) -> 33.51x Gene Wright's Benchmark HP-33s: 9900 HP-34c: 269 (my units) -> 36.8x Apart from the benchmarking feature, I am interested in how many counts you can get, either from trying the program in faster calculator or improving the method. For instance, on the HP-41 1 STO and then RCL 1 inside the loop is faster than the 1 in the beginning of the loop (thanks PeterP!). Gerson.

Re: sum of the first n integers - HP 30b Message #2 Posted by gene wright on 30 May 2010, 5:53 p.m., in response to message #1 by Gerson W. Barbosa Ok, two timings on the HP 30b. With TSYS OFF S: 254,826,600 N: 22,575 With TSYS ON S: 2,378,051,130 N: 68,964 Amazing little machine. :-)

Re: Buying an HP 30b Message #3 Posted by Namir on 30 May 2010, 6:11 p.m., in response to message #2 by gene wright Gene, Finally I was able to order the HP30b online (from southern France among all places). I got two machines. Will be glad to play with the new beast! Namir Edited: 31 May 2010, 3:07 a.m.

Re: sum of the first n integers - HP 30b Message #4 Posted by Gerson W. Barbosa on 30 May 2010, 6:19 p.m., in response to message #2 by gene wright Quote: S: 2,378,051,130 N: 68,964 That's almost 700 times more numbers than in the original problem! Well, considering the story is true which after reading the American Scientist article I am not sure about anymore. Anyway, as the Italian say, se non è vera, è ben trovata. Quote: Amazing little machine. :-) Agreed! It's a shame a scientific version is still not available.

Re: sum of the first n integers - HP 30b Message #5 Posted by gene wright on 30 May 2010, 6:28 p.m., in response to message #4 by Gerson W. Barbosa Well, Gerson... given its speed AND the scientific functions built-into it as well as being RPN and having a (relatively) low cost, it can certainly substitute for many scientific wishes! I do hope a good number here will get one as they can. And, no, I get no commission. :-)

Re: sum of the first n integers - HP 30b Message #6 Posted by Gerson W. Barbosa on 30 May 2010, 7:41 p.m., in response to message #5 by gene wright I read your review of the HP 30b in Datafile V28 N6 and found it impressive. I would get myself one, but I haven't seen it around yet. Sure I don't like inverse trigs accessible through menu, as I didn't like it on the 28S, but at least they are there.

Re: sum of the first n integers - HP 30b Message #7 Posted by gene wright on 31 May 2010, 12:17 a.m., in response to message #6 by Gerson W. Barbosa Yes, the inverse trig are there and can be assigned to the keyboard at the cost of about 24 of the 290 bytes and 3 of the 10 program locations. Really not too bad to make "run" mode much more useful. Make sure to get a copy of the HP Solve newsletter now being edited by Richard Nelson. Has had a couple of articles about the 30b with perhaps more to come.

Re: sum of the first n integers Message #8 Posted by Scott Newell on 30 May 2010, 6:58 p.m., in response to message #1 by Gerson W. Barbosa 12C+, custom integer firmware, overclocked to 48 MHz. N=152,361,930 and S=11,607,078,932,843,415. (Had to use 64 bits for the accumulator on the 12C+, and I had to check my results with bc to handle the range.) unsigned long long acc=0; do { ++x; acc+=x; } while ((((*(unsigned long int*)PIOC_PDSR) & c_mask)==c_mask));

Re: sum of the first n integers Message #9 Posted by Gerson W. Barbosa on 30 May 2010, 8:09 p.m., in response to message #8 by Scott Newell Quote: N=152,361,930 and S=11,607,078,932,843,415. That's really amazing, a hard-to-break record on the 12C+! My hopes now lie on the 50g overclocked to maximum speed. Anyone? :-) I checked your value for N on the 50g in exact mode. It worked but the calculator froze afterwards, probably because the Ni-MH had gotten low. As it would not turn off I removed one of the batteries for a moment and it restarted with the memory intact.

Re: sum of the first n integers Message #10 Posted by Scott Newell on 30 May 2010, 8:31 p.m., in response to message #9 by Gerson W. Barbosa Quote: That's really amazing, a hard-to-break record on the 12C+! My hopes now lie on the 50g overclocked to maximum speed. Anyone? :-) I suspect a C or assembly coded routine on either the 50G or the Casio would easily beat it. I can't seem to get the 12C+ to run reliably at 50MHz, and those calcs should do what, 200?

Re: sum of the first n integers Message #11 Posted by Tim Wessman on 30 May 2010, 10:14 p.m., in response to message #10 by Scott Newell 50? We keep it under 36 always as there is a risk of burning it out. . . sheesh! TW

Re: sum of the first n integers Message #12 Posted by Eric Smith on 1 June 2010, 7:16 p.m., in response to message #10 by Scott Newell 200 MHz? Where did you get that? The part has a rated maximum of 36 MHz. If you exceed that by much, reads of the flash memory will become unreliable.

Re: sum of the first n integers Message #13 Posted by Gerson W. Barbosa on 1 June 2010, 7:28 p.m., in response to message #12 by Eric Smith I think he's referring to the 50g's processor, which is underclocked at 75MHz but can run at higher speeds, up to 192MHz according to Xerxes's Benchmark article. Edited: 1 June 2010, 7:32 p.m.

Re: sum of the first n integers Message #14 Posted by Scott Newell on 1 June 2010, 10:10 p.m., in response to message #12 by Eric Smith Quote: 200 MHz? Where did you get that? The part has a rated maximum of 36 MHz. If you exceed that by much, reads of the flash memory will become unreliable. I looked up the 50G and the Casio on Wikipedia, and they claimed much faster CPUs (50G: 203 MHz?, Casio: 133 MHz?), with cache. I have not bothered to run code from sram, but that might make it to 50MHz or so. I have also not tried to increase the number of flash wait states. I did look through the AT91SAM7L128 datasheet (mine may not be the latest version) abs max ratings, but couldn't find a hard limit of 36 MHz. The PLL shows a max of 47 MHz, master clock max of 39.7 MHz, and max flash operating frequency of 39.7 MHz. Considering how sloppy the PLL reference is (20-44 kHz!), I just assumed I could overclock it a bit, just for fun.

Re: sum of the first n integers in HP 35s Message #15 Posted by Pablo P (Spain) on 31 May 2010, 4:41 a.m., in response to message #1 by Gerson W. Barbosa HP 35s ---------- S: 480 690 N: 980 ---------- A001 LBL A A002 1 A003 ENTER A004 ENTER A005 1 A006 + A007 + A008 STO A A009 LASTx A010 GTO A005 It is very slow compared to HP 33s. Edited: 31 May 2010, 4:45 a.m.

Re: sum of the first n integers in HP 35s Message #16 Posted by Gerson W. Barbosa on 31 May 2010, 6:11 p.m., in response to message #15 by Pablo P (Spain) HP-33s G0001 LBL G G0002 1 G0003 ENTER G0004 STO B H0001 LBL H H0002 RCL+ B H0003 + H0004 STO A H0005 LAST x H0006 GTO H S: 11 122 686 n: 4 716 Is the extra ENTER really necessary on the 35s? On the 33s, it is not. One less instruction inside the loop causes the count to increase about 20% (6/5), but the gain should be even greater on the 35s (Courtesy of Katie Wasserman :-) http://www.hpmuseum.org/cgi-sys/cgiwrap/hpmuseum/archv017.cgi?read=119195

Re: sum of the first n integers in HP 35s Message #17 Posted by Pablo P (Spain) on 31 May 2010, 7:16 p.m., in response to message #16 by Gerson W. Barbosa Hi Gerson, the second ENTER was unnecessary. Thanks The improved program without the constant in the loop: HP 35s ---------- S: 1 813 560 N: 1 904 ---------- A001 LBL A A002 1 A003 STO B A004 ENTER A005 RCL+ B A006 + A007 STO A A008 LASTx A009 GTO A005 Edited: 31 May 2010, 7:16 p.m.

A faster program for the HP33s Message #18 Posted by Palmer O. Hanson, Jr. on 31 May 2010, 11:23 p.m., in response to message #1 by Gerson W. Barbosa Gerson: Try this on the HP-33s starting with LASTx cleared by entering 0 X: LBL A LASTx 1 + + GTO A Read the sum from the display, rolling down if necessary. Read n by pressing LASTx, or as one less than the number in the display if the content of LASTx is 1. To cross check calculate n(n+1)/2. Two one minute tries with this method yielded Sum = 9,114,315 with n + 4,269 Sum + 9,097,245 with n + 4,265 Palmer

Re: A faster program for the HP33s Message #19 Posted by Gerson W. Barbosa on 1 June 2010, 7:19 p.m., in response to message #18 by Palmer O. Hanson, Jr. Hello Palmer, Quote: Try this on the HP-33s starting with LASTx cleared by entering 0 X: ... Read the sum from the display, rolling down if necessary. Read n by pressing LASTx, or as one less than the number in the display if the content of LASTx is 1. To cross check calculate n(n+1)/2. You're right, I should have sticked to the old K.I.S.S. rule :-) Your HP-33s is a bit faster than mine. I obtained 4245 and 4254. The slightly modified version below is even faster, provided 1 is previously stored in register B: A0001 LBL A A0002 LASTx A0003 RCL+ B A0004 + A0005 GTO A time (s) n S 59 + 99/100 5 255 13 810 140 59 + 97/100 5 252 13 794 378 Thanks for your suggestion, Gerson.

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