Q:

Is there a special name for the inverse of the simple function?

Is there a special name for the function $y (x)$ given by the equation
$$
y-\sin (y)=x?
$$
What is the simplest way to compute the value of the function for a given argument?

A:

What you are looking at is the inverse of the function $f(x) = x - \sin(x)$. Since this function is strictly increasing we can use a simple binary search to compute its inverse.
It's fairly easy to see that $x - 1 \leq f^{-1}(x) \leq x + 1$. So we initialize our binary search range to $[x-1, x+1]$. We then repeatedly evaluate the middle point of our range, and check if it's bigger or smaller than our original input, and respectively replace the higher or lower portion of our range. We keep doing this until our range is smaller than a certain precision we want.
In Python:
from math import sin

def inv_xsin(x, prec=1e-7):
    lo, hi = x - 1, x + 1
    while hi - lo > prec:
        mid = (lo + hi) / 2.0
        if mid - sin(mid) > x: hi = mid
        else: lo = mid
    return mid

