if that www.long-domain-name.com URL is in a page being served from www.long-domain-name.com, then you don't even need to specify the domain. all browsers will fetch an unqualified link from the same site that it came from.
i.e. use a site-relative URL rather than absolute and replace:
doing this is a good idea anyway, as it makes it easy to move the site to a different domain name. highly recommended, almost essential, if you want to have a three-stage deployment like "dev.long-domain-name.com" -> "testing.long-domain-name.com" -> "www.long-domain-name.com"
a good rule of thumb in any web development is to avoid absolute URLs whenever possible.