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Even more pi
Wed, 01/24/2007 - 00:48 | by Mgccl
NOTE: BELIEVE IT! A EVEN FASTER Pi CALCULATOR THAN THIS ONE IS HERE!
After last release, I found Gauss-Legendre algorithm, in theory, the higher precision pi is, the faster this algorithm will be, because each time it loop though, it will create 2 times more correct digit. Compare to the old one, the new one can beat it's speed at around 500 digits. It generates 2000 digit in 8 seconds. If I can find anything faster(which is quite possible), it should be Borwein's algorithm that have Nonic convergence. I will try to implement it into PHP soon.
So this is the result of comparing the new Gauss-Legendre pi calculator and the old one:
function bcpi($precision){ $limit = ceil(log($precision)/log(2))-1; bcscale($precision+6); $a = 1; $b = bcdiv(1,bcsqrt(2)); $t = 1/4; $p = 1; while($n < $limit){ $x = bcdiv(bcadd($a,$b),2); $y = bcsqrt(bcmul($a, $b)); $t = bcsub($t, bcmul($p,bcpow(bcsub($a,$x),2))); $a = $x; $b = $y; $p = bcmul(2,$p); ++$n; } return bcdiv(bcpow(bcadd($a, $b),2),bcmul(4,$t),$precision); }
NOTE: BELIEVE IT! A EVEN FASTER Pi CALCULATOR THAN THIS ONE IS HERE!
M: 15.2892227173MB, T: 0.73740196228s, QC: 160, QT: 0.0243449211121s, RT: 0.713057041168s
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Comments
[...] Even more pi [...]
[...] Even more pi [...]
[...] Even more pi [...]
[...] Even more pi [...]
[...] think use PHP to
[...] think use PHP to calculate the 2000 places after pi in 8 seconds is fast? you haven't see anything yet. This is the 3rd last attempt of finding pi (at least till [...]
Borwein's Nonic Convergence
I followed your other examples to make a bcpi function using Borwein's Nonic Convergence. The problem I ran into was it needs many many more decimal places for the calculations. I separated the iterations from the precision for testing purposes. I set the precision to 500. After 4 iterations I could not get any better than 104 decimals. I set the precision to precision*2, then *3, etc. Even at precision*4 I could only get 212 decimals out of it. For CPU comparison:
Gauss-Legendre algorithm 1.278711 seconds.
Borwein's Quartic convergence 0.582384 seconds.
Borwein's Nonic convergence (4 iterations) 26.619726 seconds. (212 correct)
With all of that said, I used bcgetscale, bcroot (0.2) and this code:
function bcpi4($precision) { // bcscale($precision+6); bcscale($precision*4); $a = bcdiv(1,3); $r = bcdiv(bcsub(bcsqrt(3),1),2); $s = bcroot(bcsub(1,bcpow($r,3)),3); // $count = ceil(log($precision+6)/log(2))-1; $count = 4; $i = 0; while($i < $count){ $t = bcadd(1,bcmul(2,$r)); $u = bcroot(bcmul(bcmul(9,$r),bcadd(1,bcadd($r,bcpow($r,2)))),3); $v = bcadd(bcpow($t,2),bcadd(bcmul($t,$u),bcpow($u,2))); $w = bcdiv(bcmul(27,bcadd(1,bcadd($s,bcpow($s,2)))),$v); if ($i == 0) { $a = bcadd(bcmul($w,$a),bcmul($a,bcsub(1,$w))); } else { $a = bcadd(bcmul($w,$a),bcmul(bcpow(3,2*$i-1),bcsub(1,$w))); } $s = bcdiv(bcpow(bcsub(1,$r),3),bcmul(bcadd($t,bcmul(2,$u)),$v)); $r = bcroot(bcsub(1,bcpow($s,3)),3); ++$i; } return bcdiv(1,$a,$precision); }When you get to it, I hope you do a much better job with this than I did.
after some thought about
after some thought about it... I think the Nonic Convergence can't be that fast.
even though each iteration more digits are produced, each iteration costs a lot of time, especially finding the root of an number with my bcroot function is extremely slow.
fuck you
fuck you
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