双双精度浮点数作为两个双精度数之和

Question

Following papers and source code for double-double arithmetic for some time, I still can't find out how exactly a dd_real ( defined as struct dd_real { double x[2];...} ) number is split into two doubles. Say if I initialize it with a string, dd_real pi = "3.14159265358979323846264338327950"; what will be pi.x[0] and pi.xi[1] ? I need to understand it and then write a hopefully small Python function that does it.

The reason I don't just want to call into the QD library is that I'd prefer to reimplement the correct split in Python so that I send my 35-digit precision constants (given as strings) as double2 to CUDA code where it will be treated as double-double reals by the GQD library -- the only library, it seems, to deal with extended precision calculations in CUDA. That unfortunately rules out mpmath too, on Python side.

Answer 1

Say that you initialize your double double with the binary number:

1.011010101111111010101010101010000000101010110110000111011111101010010101010
  < ---                 52 binary digits         --- >< --- more digits --- >

Then one double will be 1.0110101011111110101010101010100000001010101101100001 and the other will be 1.1011111101010010101010 * 2^-53

When you add these two numbers (as reals), the sum is the initial value. The first one packs as many bits as possible in its 52-bit mantissa. The second one contains the remaining bits, with the appropriate exponent.