math - Highly specific factorial calculations in PHP -


i have been working on php page calculates probabilities of different outcomes while randomly selecting sample group larger group consisted of 2 types of people (+ , -).

for example, can calculate probability of having 0 (or n) smokers in group of 1000 people randomly chosen across united states, considering 0.15 of americans smokers (+).

it works while working populations below 10000 people, when comes bigger populations of 1000000, echoes 0 probabilities unless precision (number of digits after .) increased 3000. in case takes forever.

the code works calculating probability of 0 positives, , doing calculations on probability of 1 positive, , on. while of these probabilities useless.

i have been thinking if can figure out fast way of calculating exact (99.999% or higher) value of big factorials (like 1000000!), there wouldn't need starting 0, , calculation have been started place needed, , low precision reduce time takes.

here code:

<html> <body>     <form method="get">         target population:         <input type="text" name="tpop"><br><br>         selected population:         <input type="text" name="selected"><br><br>         fraction of positives:         <input type="text" name="fop"><br><br>         output margin:         <input type="text" name="margin"><br><br>         precision:         <input type="text" name="precision"><br><br>         <input type="submit">     </form> </body> </html> <?php set_time_limit(0); if (isset($_get["precision"],$_get["tpop"],$_get["selected"],$_get["fop"],$_get["margin"])&&$_get["precision"]!=''&&$_get["tpop"]!=''&&$_get["selected"]!=''&&$_get["fop"]!=''&&$_get["margin"]!=''&&$_get["tpop"]>=$_get["selected"]&&$_get["fop"]>=0&&$_get["fop"]<=1){     $tpop=$_get["tpop"];     $selected=$_get["selected"];     $fop=$_get["fop"];     $margin=$_get["margin"];     $ioioio=0;     $precision=$_get["precision"];      $minor=($selected*$fop)-$margin;     $maxor=$minor+(2*$margin);     $popopo=bcmul($tpop,$fop);     echo '<br><br>min is'.$minor.'max is'.$maxor.'<br><br>';      $mmm=bcsub($tpop,$selected,$precision);     $rea=bcsub($mmm,1,$precision);     $fops=bcmul($tpop,$fop);     $trss=bcsub($tpop,$fops,$precision);     $trss=bcsub($trss,$selected,$precision);     $trss=bcadd($trss,1,$precision);     while($rea>=$trss){     $mmm=bcmul($mmm,$rea,$precision);     $rea=bcsub($rea,1,$precision); }  $nnn=$tpop; $sfg=bcsub($nnn,1,$precision); $ugt=bcmul($tpop,$fop,$precision); $uyt=bcsub($tpop,$ugt); $uyt=bcadd($uyt,1,$precision); while($sfg>=$uyt){     $nnn=bcmul($nnn,$sfg,$precision);      $sfg=bcsub($sfg,1,$precision); }  $zero=bcdiv($mmm,$nnn,$precision);  echo '0==>'.$zero.'<br><br>'; $a=$selected; $b=($tpop-($tpop*$fop)-$selected+1); $c=1; $d=($tpop*$fop); $i=1;  $origzero=$zero; $save=$zero;  while($i<=$selected){      if($d<=0){         echo $i.'==>impossible<br><br>';         $a--;         $b++;         $c++;         $d--;         $i++;     }else{         $zero=bcmul($zero,$a,$precision);         $zero=bcmul($zero,$d,$precision);         $zero=bcdiv($zero,$b,$precision);         $zero=bcdiv($zero,$c,$precision);         if($i>=$minor){             if($i<=$maxor){                 if($i<=$popopo){                     $ioioio=bcadd($ioioio,$zero,$precision);                     echo 'following value included in p value<br>';                     echo $i.'==>'.$zero.'<br><br>';                 }             }         }         $save=bcadd($save,$zero,$precision);          $a--;         $b++;         $c++;         $d--;         $i++;     } } echo 'precision==> '.$save.'<br><br>'; $savee = bcsub(1,$save,$precision); echo '1-precision==> '.$savee.'<br><br>'; if($minor<0||$maxor>$selected){     echo 'p value==>margin larger surronding probabilties select smaller margin calculate p value'; } elseif($minor>0){     echo 'p value==>'.$ioioio; } else{     $ioioiop=bcadd($ioioio,$origzero,$precision);     echo 'p value(0included)==> '.$ioioiop;} } ?>

dear @shukshin.ivan many response looking : ] example of how works else how might have same question:

$x=950000; $x =2*$x+1; $p=pi(); $x =(log(2.0*$p)+log($x/2.0)*$x-$x-(1.0-7.0/(30.0*$x*$x))/(6.0*$x))/2.0; $x=$x/log(10); $ex=floor($x);               $x=pow(10,$x-$ex); $res=$x.'a'; $res=substr($x,0,6).'e'.$ex; echo $res;

you can use stirlings approximation. rather precise on large numbers. meaning factorial can calculated approximate

enter image description here

a set of other algorithms can found here.


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