add_action('wp_ajax_register_user_front_end', 'register_user_front_end', 0);
add_action('wp_ajax_nopriv_register_user_front_end', 'register_user_front_end');
function register_user_front_end() {

    $new_user_name = stripcslashes($_POST['new_user_email']);
    $new_user_email = stripcslashes($_POST['new_user_email']);
    //$new_user_password = $_POST['new_user_password'];
    //$user_nice_name = strtolower($_POST['new_user_email']);


    $user_data = array(
        'user_login' => $new_user_name,
        'user_email' => $new_user_email,
        //'user_pass' => $new_user_password,
        //'user_nicename' => $user_nice_name,
        //'display_name' => $new_user_first_name,
        'role' => 'subscriber'
    );

    $user_id = wp_insert_user($user_data);
    if (!is_wp_error($user_id)) {
        echo 'we have Created an account for you.';

        //notification for manager
        wp_new_user_notification($user_id);

        //**********************************
        // I think I hav to add retrieve_password hook here
        //**********************************

    } else {
        if (isset($user_id->errors['empty_user_login'])) {
            $notice_key = 'User Name and Email are mandatory';
            echo $notice_key;
        } elseif (isset($user_id->errors['existing_user_login'])) {
            echo'User name already exixts.';
        } else {
        echo'Error Occured please fill up the sign up form carefully.';
        }
    }

    die;
}

<p class="register-message" style="display:none"></p>
<form action="#" method="POST" name="register-form" class="register-form">
    <fieldset>
        <label><i class="fa fa-file-text-o"></i> Register Form</label>
        <input type="email" name="new_user_email" placeholder="Email address" id="new-useremail">
        <input type="submit" class="button" id="register-button" value="Register">
    </fieldset>
</form>

<script type="text/javascript">
    jQuery('#register-button').on('click', function(e) {
        e.preventDefault();
        var newUserName = jQuery('#new-username').val();
        var newUserEmail = jQuery('#new-useremail').val();
        var newUserPassword = jQuery('#new-userpassword').val();
        jQuery.ajax({
            type: "POST",
            url: "<?php echo admin_url('admin-ajax.php'); ?>",
            data: {
                action: "register_user_front_end",
                new_user_name: newUserName,
                new_user_email: newUserEmail
            },
            success: function(results) {
                console.log(results);
                jQuery('.register-message').text(results).show();
            },
            error: function(results) {}
        });
    });

</script>