WordPress - login by email address only

gistfile1.txt

/* ------------------------------------------------------------------------- *
 * (4) Add custom authentication function
/* ------------------------------------------------------------------------- */
add_filter('authenticate', function($user, $email, $password){

    //Check for empty fields
    if(empty($email) || empty ($password)){
        //create new error object and add errors to it.
        $error = new WP_Error();

        if(empty($email)){ //No email
            $error->add('empty_username', __('<strong>ERROR</strong>: Email field is empty.'));
        }
        else if(!filter_var($email, FILTER_VALIDATE_EMAIL)){ //Invalid Email
            $error->add('invalid_username', __('<strong>ERROR</strong>: Email is invalid.'));
        }

        if(empty($password)){ //No password
            $error->add('empty_password', __('<strong>ERROR</strong>: Password field is empty.'));
        }

        return $error;
    }

    //Check if user exists in WordPress database
    $user = get_user_by('email', $email);

    //bad email
    if(!$user){
        $error = new WP_Error();
        $error->add('invalid', __('<strong>ERROR</strong>: Either the email or password you entered is invalid.'));
        return $error;
    }
    else{ //check password
        if(!wp_check_password($password, $user->user_pass, $user->ID)){ //bad password
            $error = new WP_Error();
            $error->add('invalid', __('<strong>ERROR</strong>: Either the email or password you entered is invalid.'));
            return $error;
        }else{
            return $user; //passed
        }
    }
}, 20, 3);

/* ------------------------------------------------------------------------- *
 * Remove "Username" from prompt in wp-login.php
/* ------------------------------------------------------------------------- */

 function change_login_function() {
    add_filter( 'gettext', 'username_change', 20, 3 );
    function username_change( $translated_text, $text, $domain ) 
    {
        if ($text === 'Username or Email Address') 
        {
            $translated_text = 'Email Address';
        }
        return $translated_text;
    }
}
add_action( 'login_head', 'change_login_function' );

// 	Fix lost password prompt and add additional advice to the Lost Password page
function ovni_embellish_retrieve_password_message($message){
	if (($action =  $_REQUEST['action' ]) == "lostpassword"){
		$message = str_replace("username or ", "", $message);	// drop the reference to username
		$message_addition = " If you don't see the email, check your spam filter.<br>If you repeat your request for a reset link, note the time when you make the request. Use the link in the email sent at that time. The link in earlier emails will no longer be valid.</p>";
		$message = substr_replace($message,$message_addition,-4);
		}
	return $message;
	}
add_filter('login_message', 'ovni_embellish_retrieve_password_message', 1);