When I first heard of software that could do extended precision calculation, I thought it would be very useful. Years later, I haven’t had much use for it. When I’ve though I needed extended precision, I’ve usually found a better way to solve my problem using ordinary precision and a little pencil-and-paper math. Avoiding extended precision calculation has caused me to understand my problems better. (Here’s a recent example.)
I’m not saying that extended precision isn’t sometimes necessary, or that I would go to great lengths to avoid using it. I’m only saying that I’ve had little use for it, much less than I expected, and that I’ve learned a few things by not immediately resorting to brute force.
(Why 53 bits? That’s the precision of an IEEE 754 standard floating point number, regrettably called a “double.” It’s no longer “double,” it’s typical.)
Software cannot offer infinite precision, so you have to carry out your calculations to some finite precision. If the roughly 15 decimal places of standard precision is not enough, how much do you need? How about 50? Or 100? How do you know what to choose? If you hope more precision will eliminate the need to understand what’s going on numerically, good luck with that. Maybe it will. Or maybe you’ll still see the same kinds of problems you had with standard precision.
A good use of extended precision might be as follows. You’re trying to compute the difference between two numbers that agree to 30 decimal places, so 40 decimal places of precision in your calculation will give you 10 decimal places in your result. But suppose you think “This isn’t what I expect. I’ll try a little more precision.” The computer may be trying to tell you that you’re going about something the wrong way, and the extra precision could mask your problem and give you confidence in a wrong answer.