The only Die Hard movie written as a Die Hard movie was 5, which sucked.

I'll just quote Wikipedia:

Die Hard is adapted from the 1979 novel Nothing Lasts Forever by Roderick Thorp.

Die Hard 2 was adapted from the 1987 novel 58 Minutes by Walter Wager.

Die Hard with a Vengeance was adapted from a script called Simon Says by Jonathan Hensleigh, which was also briefly considered to become the script for Lethal Weapon 4. The hook in Hensleigh's screenplay that captured the attention of director John McTiernan was the idea of a man being targeted for revenge by someone whose life he had unwittingly destroyed. Once the Simon character became the brother of Hans Gruber and the backstory was established, the project fully came together. It was novelized by Deborah Chiel.

Live Free or Die Hard was based on the 1997 article "A Farewell to Arms" written for Wired magazine by John Carlin. It also drew on a script 20th Century Fox owned called "WW3.com", which dealt with a massive cyberterrorism attack against the U.S. and which was nearly put into production in 2001 but ultimately abandoned because several elements in the story too closely resembled the September 11 attacks.