Given's Rotation SVD example, simplified

Average Error: 15.9 → 0.2

Time: 23.8s

Precision: 64

Math

\[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le -0.00255834573107373806910569413730627275072:\\ \;\;\;\;\frac{\frac{{1}^{6} - {\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right)}^{3}}{\mathsf{fma}\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5, \mathsf{fma}\left(1, 1, \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right), {\left(\sqrt{1}\right)}^{8}\right)}}{1 + \frac{\sqrt{0.5 \cdot \left({1}^{3} + {\left(\frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}}{\sqrt{\mathsf{fma}\left(1, 1, \frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \left(\frac{1}{\mathsf{hypot}\left(1, x\right)} - 1\right)\right)}}}\\ \mathbf{elif}\;x \le 0.002020724333725314961901498023166823259089:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{{x}^{2}}{{\left(\sqrt{1}\right)}^{3}}, 0.25, 0.5 - \mathsf{fma}\left(\frac{{x}^{4}}{{\left(\sqrt{1}\right)}^{5}}, 0.1875, \frac{0.5}{\sqrt{1}}\right)\right)}{1 + \frac{\sqrt{0.5 \cdot \left({1}^{3} + {\left(\frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}}{\sqrt{\mathsf{fma}\left(1, 1, \frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \left(\frac{1}{\mathsf{hypot}\left(1, x\right)} - 1\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot 1 - \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5}{1 + \sqrt[3]{{\left(\frac{\sqrt{0.5 \cdot \left({1}^{3} + {\left(\frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}}{\sqrt{\mathsf{fma}\left(1, 1, \frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \left(\frac{1}{\mathsf{hypot}\left(1, x\right)} - 1\right)\right)}}\right)}^{3}}}\\ \end{array}\]

C

double f(double x) {
        double r215828 = 1.0;
        double r215829 = 0.5;
        double r215830 = x;
        double r215831 = hypot(r215828, r215830);
        double r215832 = r215828 / r215831;
        double r215833 = r215828 + r215832;
        double r215834 = r215829 * r215833;
        double r215835 = sqrt(r215834);
        double r215836 = r215828 - r215835;
        return r215836;
}

↓

double f(double x) {
        double r215837 = x;
        double r215838 = -0.002558345731073738;
        bool r215839 = r215837 <= r215838;
        double r215840 = 1.0;
        double r215841 = 6.0;
        double r215842 = pow(r215840, r215841);
        double r215843 = hypot(r215840, r215837);
        double r215844 = r215840 / r215843;
        double r215845 = r215840 + r215844;
        double r215846 = 0.5;
        double r215847 = r215845 * r215846;
        double r215848 = 3.0;
        double r215849 = pow(r215847, r215848);
        double r215850 = r215842 - r215849;
        double r215851 = fma(r215840, r215840, r215847);
        double r215852 = sqrt(r215840);
        double r215853 = 8.0;
        double r215854 = pow(r215852, r215853);
        double r215855 = fma(r215847, r215851, r215854);
        double r215856 = r215850 / r215855;
        double r215857 = pow(r215840, r215848);
        double r215858 = pow(r215844, r215848);
        double r215859 = r215857 + r215858;
        double r215860 = r215846 * r215859;
        double r215861 = sqrt(r215860);
        double r215862 = r215844 - r215840;
        double r215863 = r215844 * r215862;
        double r215864 = fma(r215840, r215840, r215863);
        double r215865 = sqrt(r215864);
        double r215866 = r215861 / r215865;
        double r215867 = r215840 + r215866;
        double r215868 = r215856 / r215867;
        double r215869 = 0.002020724333725315;
        bool r215870 = r215837 <= r215869;
        double r215871 = 2.0;
        double r215872 = pow(r215837, r215871);
        double r215873 = pow(r215852, r215848);
        double r215874 = r215872 / r215873;
        double r215875 = 0.25;
        double r215876 = 4.0;
        double r215877 = pow(r215837, r215876);
        double r215878 = 5.0;
        double r215879 = pow(r215852, r215878);
        double r215880 = r215877 / r215879;
        double r215881 = 0.1875;
        double r215882 = r215846 / r215852;
        double r215883 = fma(r215880, r215881, r215882);
        double r215884 = r215846 - r215883;
        double r215885 = fma(r215874, r215875, r215884);
        double r215886 = r215885 / r215867;
        double r215887 = r215840 * r215840;
        double r215888 = r215887 - r215847;
        double r215889 = pow(r215866, r215848);
        double r215890 = cbrt(r215889);
        double r215891 = r215840 + r215890;
        double r215892 = r215888 / r215891;
        double r215893 = r215870 ? r215886 : r215892;
        double r215894 = r215839 ? r215868 : r215893;
        return r215894;
}

Error

Bits error versus x

Derivation

1. Split input into 3 regimes

2. if x < -0.002558345731073738

   1. Initial program 1.0

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]
   2. Using strategy rm

   3. Applied flip-- 1.0

      \[\leadsto \color{blue}{\frac{1 \cdot 1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \cdot \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}}\]

   4. Simplified 0.1

      \[\leadsto \frac{\color{blue}{1 \cdot 1 - \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]
   5. Using strategy rm

   6. Applied flip3-+ 0.1

      \[\leadsto \frac{1 \cdot 1 - \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5}{1 + \sqrt{0.5 \cdot \color{blue}{\frac{{1}^{3} + {\left(\frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{1 \cdot 1 + \left(\frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)} - 1 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}}}\]

   7. Applied associate-*r/ 0.1

      \[\leadsto \frac{1 \cdot 1 - \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5}{1 + \sqrt{\color{blue}{\frac{0.5 \cdot \left({1}^{3} + {\left(\frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}{1 \cdot 1 + \left(\frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)} - 1 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}}}\]

   8. Applied sqrt-div 0.1

      \[\leadsto \frac{1 \cdot 1 - \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5}{1 + \color{blue}{\frac{\sqrt{0.5 \cdot \left({1}^{3} + {\left(\frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}}{\sqrt{1 \cdot 1 + \left(\frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)} - 1 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}}}\]

   9. Simplified 0.1

      \[\leadsto \frac{1 \cdot 1 - \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5}{1 + \frac{\sqrt{0.5 \cdot \left({1}^{3} + {\left(\frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}}{\color{blue}{\sqrt{\mathsf{fma}\left(1, 1, \frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \left(\frac{1}{\mathsf{hypot}\left(1, x\right)} - 1\right)\right)}}}}\]
   10. Using strategy rm

   11. Applied flip3-- 0.1

       \[\leadsto \frac{\color{blue}{\frac{{\left(1 \cdot 1\right)}^{3} - {\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right)}^{3}}{\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right) + \left(\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right) \cdot \left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right) + \left(1 \cdot 1\right) \cdot \left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right)\right)}}}{1 + \frac{\sqrt{0.5 \cdot \left({1}^{3} + {\left(\frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}}{\sqrt{\mathsf{fma}\left(1, 1, \frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \left(\frac{1}{\mathsf{hypot}\left(1, x\right)} - 1\right)\right)}}}\]

   12. Simplified 0.1

       \[\leadsto \frac{\frac{\color{blue}{{1}^{6} - {\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right)}^{3}}}{\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right) + \left(\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right) \cdot \left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right) + \left(1 \cdot 1\right) \cdot \left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right)\right)}}{1 + \frac{\sqrt{0.5 \cdot \left({1}^{3} + {\left(\frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}}{\sqrt{\mathsf{fma}\left(1, 1, \frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \left(\frac{1}{\mathsf{hypot}\left(1, x\right)} - 1\right)\right)}}}\]

   13. Simplified 0.1

       \[\leadsto \frac{\frac{{1}^{6} - {\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right)}^{3}}{\color{blue}{\mathsf{fma}\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5, \mathsf{fma}\left(1, 1, \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right), {\left(\sqrt{1}\right)}^{8}\right)}}}{1 + \frac{\sqrt{0.5 \cdot \left({1}^{3} + {\left(\frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}}{\sqrt{\mathsf{fma}\left(1, 1, \frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \left(\frac{1}{\mathsf{hypot}\left(1, x\right)} - 1\right)\right)}}}\]

   if -0.002558345731073738 < x < 0.002020724333725315

   1. Initial program 31.0

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]
   2. Using strategy rm

   3. Applied flip-- 31.0

      \[\leadsto \color{blue}{\frac{1 \cdot 1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \cdot \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}}\]

   4. Simplified 31.0

      \[\leadsto \frac{\color{blue}{1 \cdot 1 - \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]
   5. Using strategy rm

   6. Applied flip3-+ 31.0

      \[\leadsto \frac{1 \cdot 1 - \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5}{1 + \sqrt{0.5 \cdot \color{blue}{\frac{{1}^{3} + {\left(\frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{1 \cdot 1 + \left(\frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)} - 1 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}}}\]

   7. Applied associate-*r/ 31.0

      \[\leadsto \frac{1 \cdot 1 - \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5}{1 + \sqrt{\color{blue}{\frac{0.5 \cdot \left({1}^{3} + {\left(\frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}{1 \cdot 1 + \left(\frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)} - 1 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}}}\]

   8. Applied sqrt-div 31.0

      \[\leadsto \frac{1 \cdot 1 - \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5}{1 + \color{blue}{\frac{\sqrt{0.5 \cdot \left({1}^{3} + {\left(\frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}}{\sqrt{1 \cdot 1 + \left(\frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)} - 1 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}}}\]

   9. Simplified 31.0

      \[\leadsto \frac{1 \cdot 1 - \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5}{1 + \frac{\sqrt{0.5 \cdot \left({1}^{3} + {\left(\frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}}{\color{blue}{\sqrt{\mathsf{fma}\left(1, 1, \frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \left(\frac{1}{\mathsf{hypot}\left(1, x\right)} - 1\right)\right)}}}}\]

   10. Taylor expanded around 0 31.0

       \[\leadsto \frac{\color{blue}{\left(0.25 \cdot \frac{{x}^{2}}{{\left(\sqrt{1}\right)}^{3}} + 0.5\right) - \left(0.5 \cdot \frac{1}{\sqrt{1}} + 0.1875 \cdot \frac{{x}^{4}}{{\left(\sqrt{1}\right)}^{5}}\right)}}{1 + \frac{\sqrt{0.5 \cdot \left({1}^{3} + {\left(\frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}}{\sqrt{\mathsf{fma}\left(1, 1, \frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \left(\frac{1}{\mathsf{hypot}\left(1, x\right)} - 1\right)\right)}}}\]

   11. Simplified 0.2

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{{x}^{2}}{{\left(\sqrt{1}\right)}^{3}}, 0.25, 0.5 - \mathsf{fma}\left(\frac{{x}^{4}}{{\left(\sqrt{1}\right)}^{5}}, 0.1875, \frac{0.5}{\sqrt{1}}\right)\right)}}{1 + \frac{\sqrt{0.5 \cdot \left({1}^{3} + {\left(\frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}}{\sqrt{\mathsf{fma}\left(1, 1, \frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \left(\frac{1}{\mathsf{hypot}\left(1, x\right)} - 1\right)\right)}}}\]

   if 0.002020724333725315 < x

   1. Initial program 1.1

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]
   2. Using strategy rm

   3. Applied flip-- 1.1

      \[\leadsto \color{blue}{\frac{1 \cdot 1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \cdot \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}}\]

   4. Simplified 0.1

      \[\leadsto \frac{\color{blue}{1 \cdot 1 - \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]
   5. Using strategy rm

   6. Applied flip3-+ 0.1

      \[\leadsto \frac{1 \cdot 1 - \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5}{1 + \sqrt{0.5 \cdot \color{blue}{\frac{{1}^{3} + {\left(\frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{1 \cdot 1 + \left(\frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)} - 1 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}}}\]

   7. Applied associate-*r/ 0.1

      \[\leadsto \frac{1 \cdot 1 - \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5}{1 + \sqrt{\color{blue}{\frac{0.5 \cdot \left({1}^{3} + {\left(\frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}{1 \cdot 1 + \left(\frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)} - 1 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}}}\]

   8. Applied sqrt-div 0.1

      \[\leadsto \frac{1 \cdot 1 - \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5}{1 + \color{blue}{\frac{\sqrt{0.5 \cdot \left({1}^{3} + {\left(\frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}}{\sqrt{1 \cdot 1 + \left(\frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)} - 1 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}}}\]

   9. Simplified 0.1

      \[\leadsto \frac{1 \cdot 1 - \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5}{1 + \frac{\sqrt{0.5 \cdot \left({1}^{3} + {\left(\frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}}{\color{blue}{\sqrt{\mathsf{fma}\left(1, 1, \frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \left(\frac{1}{\mathsf{hypot}\left(1, x\right)} - 1\right)\right)}}}}\]
   10. Using strategy rm

   11. Applied add-cbrt-cube 0.1

       \[\leadsto \frac{1 \cdot 1 - \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5}{1 + \frac{\sqrt{0.5 \cdot \left({1}^{3} + {\left(\frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}}{\color{blue}{\sqrt[3]{\left(\sqrt{\mathsf{fma}\left(1, 1, \frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \left(\frac{1}{\mathsf{hypot}\left(1, x\right)} - 1\right)\right)} \cdot \sqrt{\mathsf{fma}\left(1, 1, \frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \left(\frac{1}{\mathsf{hypot}\left(1, x\right)} - 1\right)\right)}\right) \cdot \sqrt{\mathsf{fma}\left(1, 1, \frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \left(\frac{1}{\mathsf{hypot}\left(1, x\right)} - 1\right)\right)}}}}}\]

   12. Applied add-cbrt-cube 0.1

       \[\leadsto \frac{1 \cdot 1 - \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5}{1 + \frac{\color{blue}{\sqrt[3]{\left(\sqrt{0.5 \cdot \left({1}^{3} + {\left(\frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)} \cdot \sqrt{0.5 \cdot \left({1}^{3} + {\left(\frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}\right) \cdot \sqrt{0.5 \cdot \left({1}^{3} + {\left(\frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}}}}{\sqrt[3]{\left(\sqrt{\mathsf{fma}\left(1, 1, \frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \left(\frac{1}{\mathsf{hypot}\left(1, x\right)} - 1\right)\right)} \cdot \sqrt{\mathsf{fma}\left(1, 1, \frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \left(\frac{1}{\mathsf{hypot}\left(1, x\right)} - 1\right)\right)}\right) \cdot \sqrt{\mathsf{fma}\left(1, 1, \frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \left(\frac{1}{\mathsf{hypot}\left(1, x\right)} - 1\right)\right)}}}}\]

   13. Applied cbrt-undiv 0.1

       \[\leadsto \frac{1 \cdot 1 - \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5}{1 + \color{blue}{\sqrt[3]{\frac{\left(\sqrt{0.5 \cdot \left({1}^{3} + {\left(\frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)} \cdot \sqrt{0.5 \cdot \left({1}^{3} + {\left(\frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}\right) \cdot \sqrt{0.5 \cdot \left({1}^{3} + {\left(\frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}}{\left(\sqrt{\mathsf{fma}\left(1, 1, \frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \left(\frac{1}{\mathsf{hypot}\left(1, x\right)} - 1\right)\right)} \cdot \sqrt{\mathsf{fma}\left(1, 1, \frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \left(\frac{1}{\mathsf{hypot}\left(1, x\right)} - 1\right)\right)}\right) \cdot \sqrt{\mathsf{fma}\left(1, 1, \frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \left(\frac{1}{\mathsf{hypot}\left(1, x\right)} - 1\right)\right)}}}}}\]

   14. Simplified 0.1

       \[\leadsto \frac{1 \cdot 1 - \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5}{1 + \sqrt[3]{\color{blue}{{\left(\frac{\sqrt{0.5 \cdot \left({1}^{3} + {\left(\frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}}{\sqrt{\mathsf{fma}\left(1, 1, \frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \left(\frac{1}{\mathsf{hypot}\left(1, x\right)} - 1\right)\right)}}\right)}^{3}}}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.00255834573107373806910569413730627275072:\\ \;\;\;\;\frac{\frac{{1}^{6} - {\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right)}^{3}}{\mathsf{fma}\left(\left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5, \mathsf{fma}\left(1, 1, \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5\right), {\left(\sqrt{1}\right)}^{8}\right)}}{1 + \frac{\sqrt{0.5 \cdot \left({1}^{3} + {\left(\frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}}{\sqrt{\mathsf{fma}\left(1, 1, \frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \left(\frac{1}{\mathsf{hypot}\left(1, x\right)} - 1\right)\right)}}}\\ \mathbf{elif}\;x \le 0.002020724333725314961901498023166823259089:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{{x}^{2}}{{\left(\sqrt{1}\right)}^{3}}, 0.25, 0.5 - \mathsf{fma}\left(\frac{{x}^{4}}{{\left(\sqrt{1}\right)}^{5}}, 0.1875, \frac{0.5}{\sqrt{1}}\right)\right)}{1 + \frac{\sqrt{0.5 \cdot \left({1}^{3} + {\left(\frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}}{\sqrt{\mathsf{fma}\left(1, 1, \frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \left(\frac{1}{\mathsf{hypot}\left(1, x\right)} - 1\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot 1 - \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right) \cdot 0.5}{1 + \sqrt[3]{{\left(\frac{\sqrt{0.5 \cdot \left({1}^{3} + {\left(\frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}}{\sqrt{\mathsf{fma}\left(1, 1, \frac{1}{\mathsf{hypot}\left(1, x\right)} \cdot \left(\frac{1}{\mathsf{hypot}\left(1, x\right)} - 1\right)\right)}}\right)}^{3}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 +o rules:numerics
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1 (sqrt (* 0.5 (+ 1 (/ 1 (hypot 1 x)))))))