When is FFT multiplication of arbitrary-precision polynomials practical?

It is well recognized in the computer algebra theory and systems communities that the Fast Fourier Transform (FFT) can be used for multiplying polynomials. Theory predicts that it is fast for “large enough” polynomials. Even so, it appears that no major general-purpose computer algebra system uses the FFT as a general default technique for this purpose. Some provide (optional) access to FFT techniques, so it is possible that a rational decision was made against using FFT. By contrast, the implementors of some more restricted systems, typically specializing in univariate, dense polynomials over finite or infinite fields, seem to think the FFT is good, in practice, even for smallish problems, and so they have made the decision the other way. We provide some benchmarks for polynomial multiplication that the FFT should be good at, and also suggest an alternative version of FFT polynomial multiplication using floating-point numbers that has some nicer properties than the typical finite-field “Number Theoretic Transform” approach. In particular, in spite of the fact that a floating-point FFT gives approximate answers, we can nevertheless use it to produce EXACT answers to a class of polynomial multiplication problems for arbitrary-precision coefficient polynomials, simply by noting that a sufficiently-accurate coefficient, namely error is less than 0.5, we have the exact integer answer. We also have some more data on what “large enough” to warrant the use of FFT, might be.