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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 r72712710 = 1.0;
        double r72712711 = 0.5;
        double r72712712 = x;
        double r72712713 = hypot(r72712710, r72712712);
        double r72712714 = r72712710 / r72712713;
        double r72712715 = r72712710 + r72712714;
        double r72712716 = r72712711 * r72712715;
        double r72712717 = sqrt(r72712716);
        double r72712718 = r72712710 - r72712717;
        return r72712718;
}

Reproduce

herbie shell --seed 2019124 +o rules:numerics
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  (- 1 (sqrt (* 1/2 (+ 1 (/ 1 (hypot 1 x)))))))