From the group Crew Stories on Facebook:

For Gruber's fall from Nakatomi Plaza, Alan Rickman was dropped about 25 feet. He was suspended on a raised platform and dropped onto a blue screen airbag. This allowed the background behind him to be composited with footage taken from Fox Plaza and falling confetti that looked like bearer bonds. Rickman had to fall backward onto the bag, something stuntmen avoid to control their fall. Director John McTiernan convinced Rickman by demonstrating the stunt himself and falling onto a pile of cardboard boxes. Rickman was told he would be dropped on a count of three, stunt coordinator, Charlie Picerni, devised a plan to drop Rickman at "one" in order to provoke a genuine reaction of shock. McTiernan said, "there's no way he could fake that". The first take was used, but McTiernan convinced Rickman to perform a second one as backup.

Capturing the stunt was difficult because it was impossible for a human to focus the camera fast enough to prevent the image from blurring as Rickman fell away. Supervised by visual effects producer Richard Edlund, Boss Film Studios engineered an automated system using a computer that rapidly refocused the camera via a motor on its focus ring. A wide-angle lens camera shooting at 270 frames per second was used, creating footage that played 10 times slower than normal. Despite these innovations, the camera struggled to keep Rickman entirely in focus during his 1.5-second fall; the scene cuts away from Rickman as the usable footage runs out. To complete Gruber's fatal descent, Kenny Bates was lowered 318 feet (97 m) from Fox Plaza in a harness that slowed his fall as he neared the ground. Some of the Fox Plaza residents, frustrated by the debris and destruction around the building, refused to turn off their office lights for exterior shots of the Plaza.

Stunt man Kenny Bates won the Science and Technical Academy Award for the design and development of the Decelerator System, which provides two advantages. First, it allows a stuntperson to fall from much higher platforms. “To back up a little,” Mr. Bates explains, “just to give you an idea of how this came to be, if you date back into the early days of motion picture history, when stuntmen first started doing high falls, they would do it into water, or they would put up two sawhorses and put planks between the sawhorses, and they would actually jump, say, 15 or 20 feet onto these breakaway planks. These are how high falls basically originated.” As falls got higher, stuntmen began to use haystacks, nets, and cardboard boxes. “I’ve heard of stuntmen falling up to 10 stories, or 100 feet, into cardboard boxes. These boxes were actually set up in a configuration to break the fall.” Then came the airbag. “The highest high fall into an airbag is 311 feet. That’s 31 stories.

Most commonly, though, airbags are used for doing falls from, oh, 20 feet up to 150. The most common falls are between 20 feet and 80 feet.” While airbags are great and they’re still in use today, they still leave one problem. Shooting down. With any of these devices, the director must always shoot from the bottom up to avoid filming whatever it is the stuntman is going to land on. What’s where the Decelerator’s second advantage comes in. Since all you’ve got is a cable attached to the stuntman’s ankle, it doesn’t matter what direction you film in.

“When we did Die Hard, I started using a device called a Descender, to do controlled falls. In other words, we do a controlled fall from I’ve been anywhere up to 105 stories. The fall is controlled because your descending on a small cable. If the film is undercranked, it looks like you’re falling.” What Bates has done is used his knowledge of physics and film to calibrate the speed of the fall versus the degree to which the film must be undercranked. “In Die Hard, where Alan Rickman dies, falling backwards out of the building, that would have been a death defying feat. Instead we came in and packaged an illusion for Joel Silver. Since then I’ve done every one of his films.” He also doubled Bruce Willis when he leapt off the top of the building with a firehose.