is there a JavaScript implementation of the Inverse Error Function, akin to MATLAB erfinv()?

Question

is there a JavaScript implementation of the Inverse Error Function? This would implement the Gauss inverse error function. Approximations are ok.

Answer 1

Why yes. There is.

The following code uses built-in JavaScript functions and implments Abramowitz and Stegun's algorithm as described here:

    function erfinv(x){
        var z;
        var a  = 0.147;                                                   
        var the_sign_of_x;
        if(0==x) {
            the_sign_of_x = 0;
        } else if(x>0){
            the_sign_of_x = 1;
        } else {
            the_sign_of_x = -1;
        }

        if(0 != x) {
            var ln_1minus_x_sqrd = Math.log(1-x*x);
            var ln_1minusxx_by_a = ln_1minus_x_sqrd / a;
            var ln_1minusxx_by_2 = ln_1minus_x_sqrd / 2;
            var ln_etc_by2_plus2 = ln_1minusxx_by_2 + (2/(Math.PI * a));
            var first_sqrt = Math.sqrt((ln_etc_by2_plus2*ln_etc_by2_plus2)-ln_1minusxx_by_a);
            var second_sqrt = Math.sqrt(first_sqrt - ln_etc_by2_plus2);
            z = second_sqrt * the_sign_of_x;
        } else { // x is zero
            z = 0;
        }
  return z;
}

Answer 2

Here's an alternative implementation of Abramowitz and Stegun's algorithm (equivalent to ptmalcolm's answer, but more succinct and twice as fast):

function erfinv(x) {
  // maximum relative error = .00013
  const a  = 0.147  
  //if (0 == x) { return 0 }
  const b = 2/(Math.PI * a) + Math.log(1-x**2)/2
  const sqrt1 = Math.sqrt( b**2 - Math.log(1-x**2)/a )
  const sqrt2 = Math.sqrt( sqrt1 - b )
  return sqrt2 * Math.sign(x)
}

You can test the speed with console.time("erfinv"); for (let i=0; i<1000000000; i++) {erfinv(i/1000000000)}; console.timeEnd("erfinv")

The if statement optimization is commented out as it doesn't seem to make a difference - presumably the interpreter recognizes that this is all one equation.

If you need a more accurate approximation, check out Wikipedia.