|Welcome to the Monastery|
One possible application would be for programs that have to deal with very wild ranges of values, without loss of any precision. The example I have in mind is programs such as the Kerbal Space Program:
The KSP simulation fudges the calculations around the N-body problem in order to ensure that the simulation can be run at a pace that makes the game play possible.
Performing those calculations using your lazy lists would be so slow that it would make the game impossible to play.
And there is one reason why no one uses such representations.
Another is: how would you ever decide when to stop calling for more digits...
Floating point reals are used to calculate real trajectories; like those that took men to the moon and the Voyager missions on their grand tour passed Jupiter, Saturn, Uranus and Neptune; and to allow the Deep Space Network to still find them and receive their signals after 35 years and 10 billion miles.
Seems good enough to me.
With the rise and rise of 'Social' network sites: 'Computers are making people easier to use everyday'
Examine what is said, not who speaks -- Silence betokens consent -- Love the truth but pardon error.
"Science is about questioning the status quo. Questioning authority".
In the absence of evidence, opinion is indistinguishable from prejudice.