TITLE: How to solve Kepler's equation $M=E-\varepsilon \sin E$ for $E$?
QUESTION [5 upvotes]: I'm trying to create a program to solve a set of Kepler's Equation and I cannot isolate the single variable to use the expression in my program.
The Kepler Equation is 
$$M = E - \varepsilon   \sin(E)$$
I will enter the values of $M$ and $\varepsilon$ and wish to find the value of $E$.
The website Wolfram Alpha could find a solution for this input $30(\frac{\pi}{180})=x-0.3 \sin(x)$
How can I find $E$?
EDIT:
I would like to propose this algorithm (Javascript) to solve the equation. It might require some adjustments depending on the programming language used. I did some basic tests with it, and would like feedback on it.
The following algorithm doesn't have an user input method and is treating the expression: 
$0 = -x + \sin(-x)$
which is related to:
$x = \sin(-x)$
I used this algorithm to solve $E$ in Kepler's Equation rewriting it as (And swapping M and e for numerical constants, that in my case will be user inputs):
$0 = -M + E - e*sin(E)$
<script>
var start = Number.MIN_SAFE_INTEGER; end = Number.MAX_VALUE;

var result = null;

while(result === null) {
    var resultStart = Math.abs(-start + Math.sin(-start));
    var resultEnd = Math.abs(-end + Math.sin(-end));
    if(resultStart === 0) {
        result = start;
    } else if (resultEnd === 0) {
        result = end;
    } else {
        if (resultStart > resultEnd) {
            var startOld = start;
            start = (start+end)/2;
            if(startOld === start) { // underflow
                result = start;
            }
        } else {
            var endOld = end;
            end = (start+end)/2;
            if(endOld === end) { // underflow
                result = end;
            }
        }
    }
}

console.log(result);

</script>

REPLY [2 votes]: An analytical and elegant solution is the following Kapteyn / Fourier series:
$$E(M)=M+\sum_{n=1}\frac{ 2 J_n(n\epsilon) }{n} \sin(n M)$$
where $J_n(x)$ are the Bessel functions.
Other analytical solutions based on series expansion are discussed in:
"Solving Kepler's Equation Over Three Centuries",  Peter Colwell, 1993
For implementation of Bessel functions in Javascript, several codes are available.
For example:
http://www.mhtl.uwaterloo.ca/old/courses/me3532/js/bessel.html
https://github.com/SheetJS/bessel