How do I determine whether my calculation of pi is accurate

Question

I was trying various methods to implement a program that gives the digits of pi sequentially  I tried the Taylor series method  but it proved to converge extremely slowly  when I compared my result with the online values after some time   Anyway  I am trying better algorithms   So  while writing the program I got stuck on a problem  as with all algorithms  How do I know that the n digits that I ve calculated are accurate

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You could use multiple approaches and see if they converge to the same answer   Or grab some from the  net   The Chudnovsky algorithm is usually used as a very fast method of calculating pi  http   www craig-wood com nick articles pi-chudnovsky

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Since I m the current world record holder for the most digits of pi  I ll add my two cents   Unless you re actually setting a new world record  the common practice is just to verify the computed digits against the known values  So that s simple enough   In fact  I have a webpage that lists snippets of digits for the purpose of verifying computations against them  http   www numberworld org digits Pi     But when you get into world-record territory  there s nothing to compare against   Historically  the standard approach for verifying that computed digits are correct is to recompute the digits using a second algorithm  So if either computation goes bad  the digits at the end won t match   This does typically more than double the amount of time needed  since the second algorithm is usually slower   But it s the only way to verify the computed digits once you ve wandered into the uncharted territory of never-before-computed digits and a new world record     Back in the days where supercomputers were setting the records  two different AGM algorithms were commonly used    Gauss   Legendre algorithm Borwein s algorithm   These are both O N log N  2  algorithms that were fairly easy to implement   However  nowadays  things are a bit different  In the last three world records  instead of performing two computations  we performed only one computation using the fastest known formula  Chudnovsky Formula      This algorithm is much harder to implement  but it is a lot faster than the AGM algorithms   Then we verify the binary digits using the BBP formulas for digit extraction     This formula allows you to compute arbitrary binary digits without computing all the digits before it  So it is used to verify the last few computed binary digits  Therefore it is much faster than a full computation   The advantage of this is    Only one expensive computation is needed    The disadvantage is    An implementation of the Bailey   Borwein   Plouffe  BBP  formula is needed  An additional step is needed to verify the radix conversion from binary to decimal    I ve glossed over some details of why verifying the last few digits implies that all the digits are correct  But it is easy to see this since any computation error will propagate to the last digits     Now this last step  verifying the conversion  is actually fairly important  One of the previous world record holders actually called us out on this because  initially  I didn t give a sufficient description of how it worked   So I ve pulled this snippet from my blog   N     of decimal digits desired p   64-bit prime number     Compute A using base 10 arithmetic and B using binary arithmetic     If A   B  then with  extremely high probability   the conversion is correct     For further reading  see my blog post Pi - 5 Trillion Digits

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Undoubtedly  for your purposes  which I assume is just a programming exercise   the best thing is to check your results against any of the listings of the digits of pi on the web   And how do we know that those values are correct  Well  I could say that there are computer-science-y ways to prove that an implementation of an algorithm is correct    More pragmatically  if different people use different algorithms  and they all agree to  pick a number  a thousand  million  whatever  decimal places  that should give you a warm fuzzy feeling that they got it right   Historically  William Shanks published pi to 707 decimal places in 1873  Poor guy  he made a mistake starting at the 528th decimal place   Very interestingly  in 1995 an algorithm was published that had the property that would directly calculate the nth digit  base 16  of pi without having to calculate all the previous digits   Finally  I hope your initial algorithm wasn t pi 4   1 - 1 3   1 5 - 1 7       That may be the simplest to program  but it s also one of the slowest ways to do so  Check out the pi article on Wikipedia for faster approaches

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The Taylor series is one way to approximate pi   As noted it converges slowly   The partial sums of the Taylor series can be shown to be within some multiplier of the next term away from the true value of pi   Other means of approximating pi have similar ways to calculate the max error   We know this because we can prove it mathematically

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You could try computing sin pi 2   or cos pi 2  for that matter  using the  fairly  quickly converging power series for sin and cos    Even better  use various doubling formulas to compute nearer x 0 for faster convergence    BTW  better than using series for tan x  is  with computing say cos x  as a black box  e g  you could use taylor series as above  is to do root finding via Newton   There certainly are better algorithms out there  but if you don t want to verify tons of digits this should suffice  and it s not that tricky to implement  and you only need a bit of calculus to understand why it works

[algorithm] How do I determine whether my calculation of pi is accurate?

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