In this post I list some formulas for computing pi with elementary operations (and square root).
I'll go from worst to best. First, a horribliy bad formula that gives you three significant figures for a long computation that takes a whole second on my computer. 4*(+/%#),1>+/~*:(i.%])5e3 This also consumes too much memory so is not feasable to run for larger iteration counts, but we can modify it. The following variant needs very little memory, but still gives a very bad approximation. The increased iteration count here means this one takes about a minute and gives four significant digits. a(a=.2*+/%#)@(1>s&+)"0 s=.*:(i.%])5e4 Now a very slowly converging infinite product. This is such a bad approximation it gives you only six significant digits of precision even with a million terms. For explanation, see "http://en.wikipedia.org/wiki/Wallis_product";. +:*/*/1 1 _1 _1^~2 2 1 3+/+:i.1e6 Next, a sum that converges slowly so it's not very suitable to compute pi for precisions higher than machine floats, but that is sufficiently fast for machine precision. A hundred thousand iterations give you double precision, and a hundred iterations already give you six significant digits. This formula has a special place in my heart because I experimented with it on a programmable calculator when I was young. For explanation, see "http://en.wikipedia.org/wiki/Riemann_zeta_function";. 4%:90*+/_4^~>:i.1e5 We now go to formulas that would be reasonably scalable below machine float precision. These two compute sine and cosine using a Taylor series and use Newton's iteration to converge to pi using that. 2*(+[:(%~-.)/[:-/_2]\*/\@:%&(>:i.22))^:6]1 3*(+[:(%~0.5&-)/[:-/_2]\*/\@:%&(>:i.18))^:5]1 Lastly, an efficient and simple computation using the Taylor series of the arctangent function at tan(pi/12). The two frets below are based on the same computation, they're only written differently. (($&0 12 0 _12%i.)28)p.2-%:3 12*-/(2-%:3)(^%])1+2*i.14 The homework to the reader is to implement Plouffe's formula, or other interesting methods to compute pi. Ambrus ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
