Dynamically set input type to file

Question

How can we make the following work when we set the input type to file? Here is a fiddle too.

<div ng-app>
    <input ng-model="type" type="radio" value="email" />Email
    <input ng-model="type" type="radio" value="password" />Password
    <input ng-model="type" type="radio" value="file" />File<br/>
        <label>Type:</label> {{ type }}<br/>
    <input type="{{ type }}">
</div>

We can switch dynamically between password, email, and text input types but not into a file type.

We have also tried to set the input type to file from a directive. That also doesn't work.

There is probably a time in the DOM rendering lifecycle after which we can no longer set an input type to file. Can angular set the input type to file before it's to late?