Allow login by e-mail address

authenticate-by-email.php

<?php
/**
 * Allows customer to log in using e-mail address
 *
 * @param string username
 */

function prefix_authenticate_by_email( &$username ) {

	error_log( $username );

	$user = get_user_by( 'email', $username );

	if ( ! empty( $user ) ) :
		$username = $user->user_login;
	endif;

	return;
}
add_action( 'wp_authenticate', 'prefix_authenticate_by_email' );