Sed Command – Understanding Greediness in Regex

I want

ereg ($rat, $dog, $cat)

to become

preg_match ('#'.$rat.'#', $dog, $cat)

To achieve this, I did

echo 'ereg ($rat, $dog, $cat)' | sed "s/ereg\(.*\)(\(.*\),/preg_match\1('#'.\2.'#',/g"

but, this regex changed the

ereg ($rat, $dog, $cat)

into

preg_match ('#'.$rat, $dog.'#', $cat)

instead of

preg_match ('#'.$rat.'#', $dog, $cat)

Can someone help me to build a regex that changes

ereg ($rat, $dog, $cat)

into

preg_match ('#'.$rat.'#', $dog, $cat)