Hacker GeoHot has released ActiveRoot tool which is an APK file and user just need to install it on Samsung Galaxy S4 to root the device easily.
Root Samsung Galaxy S4 using ActiveRoot
Visit http://geohot.com/activeroot/ from your Samsung Galaxy S4 and click on lamba symbol on the web page to download and install the APK
Run installed ActiveRoot APK and wait until a message appears which says your Galaxy S4 is rooted
So, just try ActiveRoot, a free tool to root Samsung Galaxy S4 and share your experiences in comments section.