cube_root.html

<script language="JavaScript" type="text/javascript" src="BigInteger.js"></script>

<script>
function croot(cube){
    string = cube.toString();

    var cube_array = [];
    for (i=string.length; i>0; i = i-3) {
        if(i<3){
            cube_array.push(parseInt(string.substr(0, i))); //from null to i
            break;
        }
        cube_array.push(parseInt(string.substr(i-3, 3)));
    }
    cube_array = cube_array.reverse();    //invert array
    var cubes = [0, 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000]; //3 digits cubes

    var key = 0;
    var root3 = bigInt('0');
        for(i=0; i<=cubes.length; i++){
            key1 = cubes.indexOf(cubes[i]);    
            if((cubes[key1]-cube_array[key])<=0){continue;}
            else{
                key1 = key1-1;
                var korx = bigInt(key1); //cube root in the index.
                break;
            }
        }
            difference = bigInt(cube_array[key]-cubes[key1]);
            root3 = bigInt(root3.toString()+korx.toString());

    var b = true;
    var c = true;
    while(c==true){
        while(b==true)
        {
            key = key + 1;
            if(cube_array[key]===undefined){
                return root3;
            }
            subtracted = difference.multiply(bigInt('1000')).add(bigInt(cube_array[key]));
            
            //(10m+n)^3 = 1000m^3 + 300m^2n + 30mn^2 + n^3
            //(10m+n)^3 - 1000m^3 = 300m^2n + 30mn^2 + n^3
            //300m^2n + 30mn^2 + n^3     =     3(10×m)^2×n + 3(10×m)×n^2 + n^3     =     (10m+n)^3 - 1000n^3
            //n -> from 10 to 0 xecking...

            var n = bigInt('10');
            var m = root3;
            b = false; c = true;
        }
        
        //3(10×m)^2×n + 3(10×m)×n^2 + n^3
        m10 = m.multiply(bigInt('10'));             //10×m
        m102 = m10.multiply(m10);                   //(10×m)^2
        m1023 = m102.multiply(bigInt('3'));         //3(10×m)^2
        m1023n = m1023.multiply(n);                 //3(10×m)^2n
        m103 = m10.multiply(bigInt('3'));           //3(10×m)
        n2 = n.multiply(n);                         //n^2
        m103n2 = m103.multiply(n2);                 //3(10×m)×n^2
        n3 = n2.multiply(n);                        //n^3
        m1023nplm103n2 = m1023n.add(m103n2);        //3(10×m)^2×n + 3(10×m)×n^2
        m1023nplm103n2pln3 = m1023nplm103n2.add(n3);//3(10×m)^2×n + 3(10×m)×n^2 + n^3 = 300m^2n + 30mn^2 + n^3

        difference = subtracted.subtract(m1023nplm103n2pln3);

        c = false;

        if(difference.compare(bigInt('0'))=='-1'){
            n = n.subtract(bigInt('1')); //n--
            c = true; b = false;
        }
        else{
            root3 = root3.toString()+n.toString();
            root3 = bigInt(root3);
            c = true; b = true;
        }
    }
}

var x = bigInt.randBetween("1e64","2e64"); //x
var x2 = x.multiply(x);                    //x^2
var xcube = x2.multiply(x);                //x^3
document.write(
    '<br>x = '+x.toString(),                //toString()
    '<br>x<sup>2</sup> = '+x2.toString(),
    '<br>x<sup>3</sup> = '+xcube.toString()
);

var root3 = croot(xcube);

//checking
if(root3.compare(x)=='0')//bigInteger Compare
{document.write('<br><br><sup>3</sup>√(x<sup>3</sup>) = '+root3.toString()+' === x');}

</script>