You can port an existing implementation, like this one in C, to Javascript. That code has two variants, an iterative one that is more accurate and a non-interative one.
Ken Turkowski's implementation relies on splitting up the radicand into mantissa and exponent and then reassembling it, but this is only used to bring it into the range between 1/8 and 1 for the first approximation by enforcing a binary exponent between -2 and 0. In Javascript, you can do this by repeatedly dividing or multiplying by 8, which should not affect accuracy, because it is just an exponent shift.
The implementation as shown in the paper is accurate for single-precision floating-point numbers, but Javascript uses double-precision numbers. Adding two more Newton iterations yields good accuracy.
Here's the Javascript port of the described cbrt
algorithm:
Math.cbrt = function(x)
{
if (x == 0) return 0;
if (x < 0) return -Math.cbrt(-x);
var r = x;
var ex = 0;
while (r < 0.125) { r *= 8; ex--; }
while (r > 1.0) { r *= 0.125; ex++; }
r = (-0.46946116 * r + 1.072302) * r + 0.3812513;
while (ex < 0) { r *= 0.5; ex++; }
while (ex > 0) { r *= 2; ex--; }
r = (2.0 / 3.0) * r + (1.0 / 3.0) * x / (r * r);
r = (2.0 / 3.0) * r + (1.0 / 3.0) * x / (r * r);
r = (2.0 / 3.0) * r + (1.0 / 3.0) * x / (r * r);
r = (2.0 / 3.0) * r + (1.0 / 3.0) * x / (r * r);
return r;
}
I haven't tested it extensively, especially not in badly defined corner cases, but the tests and comparisons with pow
I have done look okay. Performance is probably not so great.