Databases
StackOverflow2013
Postsdbo
PO
 

Note that there are some explanatory texts on larger screens.

plurals
  1. PO
    primarykey
    data
    text
    <p>The <a href="http://en.wikipedia.org/wiki/Monte_Carlo_method" rel="noreferrer">Monte Carlo method</a>, as mentioned, applies some great concepts but it is, clearly, not the fastest, not by a long shot, not by any reasonable measure. Also, it all depends on what kind of accuracy you are looking for. The fastest π I know of is the one with the digits hard coded. Looking at <a href="http://functions.wolfram.com/Constants/Pi/" rel="noreferrer" title="pi">Pi</a> and <a href="http://functions.wolfram.com/PDF/Pi.pdf" rel="noreferrer" title="pi formulas">Pi[PDF]</a>, there are a lot of formulae.</p> <p>Here is a method that converges quickly — about 14 digits per iteration. <a href="http://numbers.computation.free.fr/Constants/PiProgram/pifast.html" rel="noreferrer" title="PiFast">PiFast</a>, the current fastest application, uses this formula with the <a href="http://www.ele.uri.edu/~hansenj/projects/ele436/fft.pdf" rel="noreferrer">FFT</a>. I'll just write the formula, since the code is straightforward. This formula was almost found by <a href="http://numbers.computation.free.fr/Constants/Pi/piramanujan.html" rel="noreferrer">Ramanujan and discovered by Chudnovsky</a>. It is actually how he calculated several billion digits of the number — so it isn't a method to disregard. The formula will overflow quickly and, since we are dividing factorials, it would be advantageous then to delay such calculations to remove terms.</p> <p><img src="https://i.stack.imgur.com/aQMkk.gif" alt="enter image description here"></p> <p><img src="https://i.stack.imgur.com/2y2l9.gif" alt="enter image description here"></p> <p>where,</p> <p><img src="https://i.stack.imgur.com/QqVnB.gif" alt="enter image description here"></p> <p>Below is the <a href="http://mathworld.wolfram.com/Brent-SalaminFormula.html" rel="noreferrer" title="Brent Salamin Formula">Brent–Salamin algorithm</a>. Wikipedia mentions that when <strong>a</strong> and <strong>b</strong> are "close enough" then <strong>(a + b)² / 4t</strong> will be an approximation of π. I'm not sure what "close enough" means, but from my tests, one iteration got 2 digits, two got 7, and three had 15, of course this is with doubles, so it might have an error based on its representation and the <em>true</em> calculation could be more accurate.</p> <pre><code>let pi_2 iters = let rec loop_ a b t p i = if i = 0 then a,b,t,p else let a_n = (a +. b) /. 2.0 and b_n = sqrt (a*.b) and p_n = 2.0 *. p in let t_n = t -. (p *. (a -. a_n) *. (a -. a_n)) in loop_ a_n b_n t_n p_n (i - 1) in let a,b,t,p = loop_ (1.0) (1.0 /. (sqrt 2.0)) (1.0/.4.0) (1.0) iters in (a +. b) *. (a +. b) /. (4.0 *. t) </code></pre> <p>Lastly, how about some pi golf (800 digits)? 160 characters!</p> <pre><code>int a=10000,b,c=2800,d,e,f[2801],g;main(){for(;b-c;)f[b++]=a/5;for(;d=0,g=c*2;c-=14,printf("%.4d",e+d/a),e=d%a)for(b=c;d+=f[b]*a,f[b]=d%--g,d/=g--,--b;d*=b);} </code></pre>
    singulars
    1. This table or related slice is empty.
    1. POWhat is the fastest way to get the value of π?
      singulars
      1. PTQuestion
    1. PTAnswer
    1. This table or related slice is empty.
    1. USnlucaroni
    plurals
    1. This table or related slice is empty.
    1. This table or related slice is empty.
    1. POWhat is the fastest way to get the value of π?
      singulars
      1. PTQuestion
    1. This table or related slice is empty.
    1. VO
      singulars
      1. PO
      1. This table or related slice is empty.
      1. VTUpMod
    2. VO
      singulars
      1. PO
      1. This table or related slice is empty.
      1. VTUpMod
    3. VO
      singulars
      1. PO
      1. This table or related slice is empty.
      1. VTUpMod
    1. COAssuming you are trying to implement the first one yourself, wouldn't sqr(k3) be a problem? I'm pretty sure it would end up an irrational number that you will have to estimate (IIRC, all roots that are not whole numbers are irrational). Everything else looks pretty straight-forward if you are using infinite precision arithmetic but that square root is a deal breaker. The second one includes a sqrt as well.
      singulars
      1. PO
      1. USBill K
    2. COin my experience, 'close enough' usually means there's a taylor series approximation involved.
      singulars
      1. PO
      1. USStephen
 

Querying!

 
Guidance
A row detail

Detail views are divided into sections. All the information in the data section comes from columns in the selected row. The other sections display data from other, related rows.

Related data can be related in a to-one or a to-many fashion. Captions of data related in a to-many fashion link to a list view showing a filtered view of the table.

Try moving around until you find a non-empty to-many entry and click on the label to get to one. You can move back to the root by clicking on the database name in the header.

SQuiL has stopped working due to an internal error.

If you are curious you may find further information in the browser console, which is accessible through the devtools (F12).

Reload