Why hypot() function is so slow?

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It’s not a simple sqrt function. You should check this link for the implementation of the algorithm: http://www.koders.com/c/fid7D3C8841ADC384A5F8DE0D081C88331E3909BF3A.aspx

It has while loop to check for convergence

/* Slower but safer algorithm due to Moler and Morrison.  Never
         produces any intermediate result greater than roughly the
         larger of X and Y.  Should converge to machine-precision
         accuracy in 3 iterations.  */

      double r = ratio*ratio, t, s, p = abig, q = asmall;

      do {
        t = 4. + r;
        if (t == 4.)
          break;
        s = r / t;
        p += 2. * s * p;
        q *= s;
        r = (q / p) * (q / p);
      } while (1);

EDIT (Update from J.M):

Here is the original Moler-Morrison paper, and here is a nice follow-up due to Dubrulle.

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