actually I think if you try to login and use a wrong password, there could be a link appearing like “forgot password?”
This would send you a new one via email. Since you do not need more information apart from the email/login that should work just fine.
I would however agree with the other posts, that there should not be a button on the login form but just a link, which directs you to a form which asks you for your email/username.
Due to Charles Boyungs comment I did rethink my statement and came to the conclusion, that his criticism is actually well put. I do still think though, that it is not a bad idea to give the possibility to register in the login form.
This is, because often the user is asked to login while trying to access a certain page, if the user has no account, here is the place to have him sign up.
It could work like this.
The form („Login Form“ for the screen reader) has a field for email & password to login and the default action on hitting enter is to login.
There is however a button register. Clicked the user is redirected to a different form with all necessary informations (if there are more needed apart from email & password). If the user did fill in the email and password, it gets passed to the other form.
In case you do not need more information, clicking on register would either present you with a success page if you filled in your information already or if you did not, with a form to fill in the information and just one button „register“.
This way you do have 3 forms, but you can combine them so that it is, in my opinion easy for the user and very convenient.