Better dots for infinite continued fraction

Question

My question is related to the out put of following code. I want the \ddots to start from the center of the plus sign and continue diagonally. The way it is right now is bit odd. Any help is much appreciated.

\documentclass{article}
\usepackage{amsmath}

\begin{document}
\newcommand*{\cofrac}[2]{%
  {%
    \rlap{$\dfrac{1}{\phantom{#1}}$}%
    \genfrac{}{}{0pt}{0}{}{#1+#2}%
  }%
}
\[
\sqrt{x} =~ \cfrac{1}{a_1+\cfrac{1}{a_2+\cfrac{1}{a_3+\cfrac{1}{a_4+{\ddots}}}}}
\]
\end{document}

Answer 1

Something like this?

I just placed the \ddots inside a \raisebox of -.33\height.

\documentclass{article}
\usepackage{mathtools}

\begin{document}
\newcommand*{\cofrac}[2]{%
  {%
    \mathrlap{\dfrac{1}{\phantom{#1}}}%
    \genfrac{}{}{0pt}{0}{}{#1+#2}%
  }%
}
\[
\sqrt{x} =~ \cfrac{1}{a_1+\cfrac{1}{a_2+\cfrac{1}{a_3+\cfrac{1}{a_4+\raisebox{-.33\height}{\(\ddots\)}}}}}
\]
\end{document}