dfmul_vf2_vf_vf， WTF

[2A5F]

static INLINE CONST VECTOR_CC vfloat2 dfmul_vf2_vf_vf(vfloat x, vfloat y) {
  vfloat s = vmul_vf_vf_vf(x, y); // x * y
  return vf2setxy_vf2_vf_vf(s, 
    vfmapn_vf_vf_vf_vf(x, y, s) // x * y - s
  );
  // x * y - x * y
  // 0 ???
}

[shibatch]

Hello,

There are explanations of how to do basic double-double calculations in many places, but see, for example, page 5 of the following paper.

https://www.davidhbailey.com/dhbpapers/qd.pdf

[2A5F]

But this is vf * vf -> vf2, and the result is { x * y, 0 } anyway. What is the significance of this operation?

[blapie]

The result isn't {x * y, 0}, you have to take into account that operations are done in finite precision arithmetic.
Changing your notations slightly and writing x * y the result of the exact/infinite precision multiplication and fl(x * y) the result of the floating point one, then vf2 contains { fl(x * y), e }, which is on one hand the result of the finite precision multiplication, and on the other hand an estimation of the error e = x * y - fl(x * y) (|e| < x * y * eps).
Storing this error prevents cancellation in allowing you to re-inject the error into future operations instead, thus creating "error-free" transformations (not entirely free but at least in the first or second order).