20141229, 02:23  #23 
Nov 2007
Halifax, Nova Scotia
38_{16} Posts 
The doublecheck for π(2^{86}) completed; the result was consistent with the value I posted on Dec. 17.
Regarding timing and resource usage, I am going to release that information when I write up this work for publication. I don't plan to run calculations for other bases, because there is no limit to how many such calculations one can do. I computed π(10^{26}) because the powers of ten have been the most competitive historically. I computed π(2^{m}) for m ≤ 86 because I find two a natural base, and because there were some other powers of two for large heights available to which I could compare. This served as an additional opportunity to check my software. Thank you everyone for being interested in these calculations! 
20141229, 04:27  #24 
Aug 2006
13541_{8} Posts 

20141229, 06:00  #25  
Nov 2007
Halifax, Nova Scotia
2^{3}·7 Posts 
Quote:


20141230, 13:02  #26 
Jan 2005
Minsk, Belarus
110010000_{2} Posts 
Any thoughts about 10^27?

20150309, 01:37  #27 
Sep 2002
Database er0rr
E5C_{16} Posts 
Douglas' paper is out: http://arxiv.org/abs/1503.01839

20150309, 02:17  #28  
Nov 2007
Halifax, Nova Scotia
2^{3}×7 Posts 
Quote:


20150309, 11:45  #29  
Jan 2008
France
547 Posts 
Though I didn't read all of the article yet, here are some comments.
You don't define what a "node" is. The first occurrence of "node" page 6 talks about 16 threads, and given that you talk about 2 8core CPUs in Appendix A, I assume it's a set of cores sharing memory. Did you try to play with hyperthreading? On my 4770K (4 cores, 8 threads with HT), using Walish library (from last December) I get 1.5s for 10^15 with 12 threads while I only get 2.2s with 4 threads. It seems Walisch implementation is about twice faster than yours (accounting for threads and frequency difference); I guess this is due to your memory usage improvements? I found a missing word on page 2 line +4: Quote:


20150309, 13:29  #30  
Nov 2007
Halifax, Nova Scotia
2^{3}×7 Posts 
Quote:
Re: hyperthreading, my code is also significantly faster with hyperthreading enabled, to a similar degree as what you're seeing with the implementation from Kim Walisch. Unfortunately hyperthreading is not enabled on one of the best clusters in Canada, where I ran my calculations. I've spoken to the people in charge and they're planning to enable it some time this year, which will take some time on ~1000 compute nodes. I also noticed that the implementation by Kim Walisch is faster than mine. I think their implementation is just more numerically efficient than mine. I wrote the bulk of my pi(x) software while I was still working as a physicist. I improved my optimization skills a lot while writing that code, and am now working fulltime as a software engineer. If you've looked at the sieve of Eratosthenes implementation by Kim Walisch, you'll see that it is incredibly efficient. I think Kim Walisch and Tomás Oliveira e Silva are two of the most efficient lowlevel coders I've ever seen. I'm glad I didn't know about the implementation by Kim Walisch, or it would have interfered with my project. However, I will probably go back now and read through their code and see what I can learn. It's a bit interesting to me that I managed to "beat" Kim Walisch to pi(10^26) despite the fact that they are apparently a more efficient lowlevel coder (for now). My feeling is that they (1) were probably missing the algorithm for betweennode (distributed memory) parallelism and (2) had more trouble getting the memory usage down. I'm going to send them an email here shortly. Sorry that was a bit longwinded. Please send me any more corrections you find. 

20150310, 04:51  #31 
Aug 2006
3^{2}·5·7·19 Posts 
Great paper! Two small points on p. 1: I'd write "limited to time complexity " rather than "limited to time complexity "  even though people would probably understand it the other way, you might as well be correct. And surely the complexity of Meissel's method was for some small , not just any ?

20150310, 08:27  #32 
"Robert Gerbicz"
Oct 2005
Hungary
2^{2}·367 Posts 
Yes, that couldn't be for any eps>0 (let eps=2).
Last fiddled with by R. Gerbicz on 20150310 at 08:32 
20150310, 19:10  #33  
Nov 2007
Halifax, Nova Scotia
2^{3}×7 Posts 
Quote:
Quote:


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