$_POSTing to and modifying the WP search results page

I have a site which uses post-name permalinks. If I use the built in search I am taken to the url

http://ift.tt/15XzNK0

I have a form which I am using as a custom search popup built into a page

    $url = '/search';

    <form id="signup" method="post" action="' . $url . '">
        <input name="search_input" type="text">
        <button type="submit">Search</button>
    </form>

I want to use this form and the default search to go to the same page http://ift.tt/15XzNK0

I cannot figure out how to $_POST to whatever url wordpress uses to display it's results.