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\[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]
1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}
double f(double x) {
        double r25922729 = 1.0;
        double r25922730 = 0.5;
        double r25922731 = x;
        double r25922732 = hypot(r25922729, r25922731);
        double r25922733 = r25922729 / r25922732;
        double r25922734 = r25922729 + r25922733;
        double r25922735 = r25922730 * r25922734;
        double r25922736 = sqrt(r25922735);
        double r25922737 = r25922729 - r25922736;
        return r25922737;
}

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))