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\[1 - \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]
1 - \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}
double f(double x) {
        double r64418770 = 1.0;
        double r64418771 = 0.5;
        double r64418772 = x;
        double r64418773 = hypot(r64418770, r64418772);
        double r64418774 = r64418770 / r64418773;
        double r64418775 = r64418770 + r64418774;
        double r64418776 = r64418771 * r64418775;
        double r64418777 = sqrt(r64418776);
        double r64418778 = r64418770 - r64418777;
        return r64418778;
}

Reproduce

herbie shell --seed 2019121 +o rules:numerics
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  (- 1 (sqrt (* 1/2 (+ 1 (/ 1 (hypot 1 x)))))))