Nginx – real solution for `No input file specified.` (nginx, fpm)

Most answers to this question are, set fastcgi_param SCRIPT_FILENAME and it will work (italic formatting is broken?!).

I have set this variable (correct) but it still shows the error instead of a 404 page, because the root of the problem is here:

location ~ \.php$ {
            fastcgi_split_path_info ^(.+\.php)(/.+)$;
            fastcgi_pass unix:/var/run/php5-fpm.sock;
            fastcgi_index index.php;
            include fastcgi_params;
    }

A nonexistent path is passed to php5-fpm, this in return prints the error, which in the log looks like:

FastCGI sent in stderr:
"Unable to open primary script: ... (No such file or directory)"
while reading response header from upstream

So before the line fastcgi_pass there must be a condition to check whether the file really exists, or, if the fpm worker returns "file not found", to guide nginx to return a 404 page.

How can I do this?