Validating A Form With Options

I am helping someone out with a form. He wants the name required, easy enough, but then only an email or street address required. I have tried a couple of things and it is not working.

I am trying a very simple code:
<script type="text/javascript">
<!--

function validate_form ( )
{
valid = true;

if ( document.contact_form.name_mailing.value == "" )
{
alert ( "Please fill in the 'Your Name' box." );
valid = false;
}

if ( document.contact_form.address_mailing.value == "" || document.contact_form.email_address.value == "")
{
alert ( "Please fill in the 'Your Information' box." );
valid = false;
}

return valid;
}

//-->
</script>

Am I not using the or operator correctly? I only want to have the address or email required, not necessarily both.