Remove `product` from WooCommerce product links

// allow product links w/o `product/` base for wc
// additionaly you can add
//     RedirectMatch 301 ^/product/(.*)$ /$1
// to `.htaccess` to handle `page?foo=bar` queries
// this script is based on this article by Ryan Sechrest:
// http://ryansechrest.com/2013/04/remove-post-type-slug-in-custom-post-type-url-and-move-subpages-to-website-root-in-wordpress/

add_action(
    'pre_get_posts',
    'custom_pre_get_posts'
);

function custom_pre_get_posts($query) {
  global $wpdb;

  if(!$query->is_main_query()) {
      return;
  }

  $post_name = $query->get('name');
  $post_type = $wpdb->get_var(
    $wpdb->prepare(
      'SELECT post_type FROM ' . $wpdb->posts . ' WHERE post_name = %s LIMIT 1',
      $post_name
    )
  );

  switch($post_type) {
    case 'product':
      //$query->set('product', $post_name);
      //$query->set('post_type', $post_type);
      $query->set('post_type', 'product');
      $query->is_single = true;
      $query->is_page = false;
    break;
  }
}