gistfile1.txt

function search_redirect_form() {
  if (is_search() && ! empty( $_GET['simple'] ) ) {
        if (isset($_GET['simple']) && $_GET['simple'] == 'dogpile') {
            $query =  sanitize_text_field($_GET["s"]);
            wp_redirect(esc_url('theurl1'. $query));
            exit(); 
        }
        if (isset($_GET['simple']) && $_GET['simple'] == 'altavista') {
            $query =  sanitize_text_field($_GET["s"]);
            wp_redirect(esc_url('url2'. $query .'/field/all/mode/all/conn/and/order/nosort'));
            exit(); 
        }
        if (isset($_GET['simple']) && $_GET['simple'] == 'askjeeves') {
            $query =  sanitize_text_field($_GET["s"]);
            wp_redirect(esc_url('url3' . $query));
            exit();  
        }
        else { 
            $query =  sanitize_text_field($_GET["s"]);
            wp_redirect(esc_url($home_url . "?s=" . $query));
            exit();  
        }	
  }
}
add_action( 'pre_get_posts', 'search_redirect_form', 1 );

(yes, I Know I should rename the 'simple' variable to something a lot more clearer...)