Fast arbitrary precision logarithms with bcmath

Question

Here is what i have

function bcln($n, $scale=10) {
    $iscale = $scale+3;
    $result = '0.0';
    $i = 0;

    do {
        $pow = (1 + (2 * $i++));
        $mul = bcdiv('1', $pow, $iscale);
        $fraction = bcmul($mul, bcpow(bcsub($n, '1', $iscale) / bcadd($n, '1.0', $iscale), $pow, $iscale), $iscale);
        $lastResult = $result;
        $result = bcadd($fraction, $result, $iscale);
    } while($result !== $lastResult);

    return bcmul('2', $result, $scale);
}

But bcln(100)it takes 5.7 seconds to complete (natural log 100, 10 decimal places). In addition, it is not always more accurate for more decimal places. Is there a better algorithm?

For this particular run, 573 iterations are required to determine the result.

Answer 1

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function bclog10($n){
    //←Might need to implement some validation logic here!
    $pos=strpos($n,'.');
    if($pos===false){
        $dec_frac='.'.substr($n,0,15);$pos=strlen($n);
    }else{  $dec_frac='.'.substr(substr($n,0,$pos).substr($n,$pos+1),0,15);
    }
    return log10((float)$dec_frac)+(float)$pos;
}

, :

function bclogn($n,$base=M_E){//$base should be float: default is e
    return bclog10($n)*log(10)/log($base);
}

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: ...

function bclog10($n){//By Matthew Slyman @aaabit.com
    $m=array();// ↓ Validation, matching/processing regex…
    preg_match('/^(-)?0*([1-9][0-9]*)?(\.(0*))?([1-9][0-9]*)?([Ee](-)?0*([1-9][0-9]*))?$/',$n,$m);
    if(!isset($m[1])){throw new \Exception('Argument: not decimal number string!');}
    $sgn=$m[1];if($sgn==='-'){throw new \Exception('Cannot compute: log(<⁺0)!');}
    $abs=$m[2];$pos=strlen($abs);
    if(isset($m[4])){$fre=$m[4];}else{$fre='';}$neg=strlen($fre);
    if(isset($m[5])){$frc=$m[5];}else{$frc='';}
    if(isset($m[7])){$esgn=$m[7]==='-'?-1:1;}else{$esgn=1;}
    if(isset($m[8])){$eexp=$m[8];}else{$eexp=0;}
    if($pos===0){
        $dec_frac='.'.substr($frc,0,15);$pos=-1*$neg;
    }else{  $dec_frac='.'.substr($abs.$fre.$frc,0,15);
    }
    return log10((float)$dec_frac)+(float)$pos+($esgn*$eexp);
}