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Comment by pklausler | Hacker News Reader

Why not use the "double-double" representation, in which one extends a regular double precision value with another double that estimates its error (and then maybe a third that estimates its error, and so on...)? Double-double operations are fast in terms of floating-point operations, conversions to other floating types are free, and the precision (107.5 bits asymptotically) is nearly as good as 128-bit (112 bits).

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Sure, but:

1) As soon as you wrap it in a Python class, much of that speed advantage goes away.

2) Double-double doesn't solve the problem with the limited exponent range of doubles.

3) When the goal is to allow setting the precision to any number of bits you like, having a separate implementation for X bits complicates things a lot, especially when the X-bit arithmetic doesn't quite behave like Y-bit arithmetic for X != Y (double-double behaves somewhat differently from 108-bit floating-point).

IMO, double-double would be most interesting as a NumPy extension.

Fair enough, although I've always (well, since 1980) been skeptical that an exponent range from 10E-307 to 10E+308 fails to suffice for any real problem. I mean, come on, there's only something like 10E+80 atoms in the universe.

Very large or small numbers are useful in combinatorics and number theory. It can also happen that they are needed for intermediate steps in an algorithm, even when the inputs and outputs are moderate (the same reason that 15 digits sometimes isn't enough even when you just want 3 digits of output). Granted, you can generally work around exponent range limitations using tricks (scaling values properly or introducing logarithms), but that can obviously be a hassle.

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