3frac (problem 3.3.3)

Average Error: 9.9 → 0.2

Time: 4.0m

Precision: 64

Math

\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le -107.46436521468802:\\ \;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{2}{{x}^{5}} + \frac{\frac{2}{x}}{x \cdot x}\right)\\ \mathbf{elif}\;x \le 11426.70565163402:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(x - \mathsf{fma}\left(x, 2, 2\right)\right), \left(\mathsf{fma}\left(\left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right) - \frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right)\right), \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right) - \frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right)\right), \left(\left(\frac{1}{x - 1} - \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right) - \frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right)\right)\right) \cdot \frac{1}{x - 1}\right)\right)\right), \left(\mathsf{fma}\left(x, x, x\right) \cdot \mathsf{fma}\left(\left(\frac{\frac{1}{x - 1}}{x - 1}\right), \left(\frac{1}{x - 1}\right), \left(\left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right) - \frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right)\right) \cdot \left(\left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right) - \frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right)\right) \cdot \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right) - \frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right)\right)\right)\right)\right)\right)\right)}{\mathsf{fma}\left(\left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right) - \frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right)\right), \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right) - \frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right)\right), \left(\left(\frac{1}{x - 1} - \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right) - \frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right)\right)\right) \cdot \frac{1}{x - 1}\right)\right) \cdot \mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{x \cdot x}}{x}\\ \end{array}\]

C

double f(double x) {
        double r17623847 = 1.0;
        double r17623848 = x;
        double r17623849 = r17623848 + r17623847;
        double r17623850 = r17623847 / r17623849;
        double r17623851 = 2.0;
        double r17623852 = r17623851 / r17623848;
        double r17623853 = r17623850 - r17623852;
        double r17623854 = r17623848 - r17623847;
        double r17623855 = r17623847 / r17623854;
        double r17623856 = r17623853 + r17623855;
        return r17623856;
}

↓

double f(double x) {
        double r17623857 = x;
        double r17623858 = -107.46436521468802;
        bool r17623859 = r17623857 <= r17623858;
        double r17623860 = 2.0;
        double r17623861 = 7.0;
        double r17623862 = pow(r17623857, r17623861);
        double r17623863 = r17623860 / r17623862;
        double r17623864 = 5.0;
        double r17623865 = pow(r17623857, r17623864);
        double r17623866 = r17623860 / r17623865;
        double r17623867 = r17623860 / r17623857;
        double r17623868 = r17623857 * r17623857;
        double r17623869 = r17623867 / r17623868;
        double r17623870 = r17623866 + r17623869;
        double r17623871 = r17623863 + r17623870;
        double r17623872 = 11426.70565163402;
        bool r17623873 = r17623857 <= r17623872;
        double r17623874 = fma(r17623857, r17623860, r17623860);
        double r17623875 = r17623857 - r17623874;
        double r17623876 = cbrt(r17623860);
        double r17623877 = cbrt(r17623857);
        double r17623878 = r17623876 / r17623877;
        double r17623879 = r17623878 * r17623878;
        double r17623880 = r17623878 * r17623879;
        double r17623881 = r17623880 - r17623880;
        double r17623882 = 1.0;
        double r17623883 = r17623857 - r17623882;
        double r17623884 = r17623882 / r17623883;
        double r17623885 = r17623884 - r17623881;
        double r17623886 = r17623885 * r17623884;
        double r17623887 = fma(r17623881, r17623881, r17623886);
        double r17623888 = fma(r17623857, r17623857, r17623857);
        double r17623889 = r17623884 / r17623883;
        double r17623890 = r17623881 * r17623881;
        double r17623891 = r17623881 * r17623890;
        double r17623892 = fma(r17623889, r17623884, r17623891);
        double r17623893 = r17623888 * r17623892;
        double r17623894 = fma(r17623875, r17623887, r17623893);
        double r17623895 = r17623887 * r17623888;
        double r17623896 = r17623894 / r17623895;
        double r17623897 = r17623860 / r17623868;
        double r17623898 = r17623897 / r17623857;
        double r17623899 = r17623873 ? r17623896 : r17623898;
        double r17623900 = r17623859 ? r17623871 : r17623899;
        return r17623900;
}

Error

Bits error versus x

Target

Original	9.9
Target	0.3
Herbie	0.2

\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

1. Split input into 3 regimes

2. if x < -107.46436521468802

   1. Initial program 19.7

      \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]

   2. Taylor expanded around inf 0.5

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{x}^{7}} + \left(2 \cdot \frac{1}{{x}^{3}} + 2 \cdot \frac{1}{{x}^{5}}\right)}\]

   3. Simplified 0.1

      \[\leadsto \color{blue}{\frac{2}{{x}^{7}} + \left(\frac{\frac{2}{x}}{x \cdot x} + \frac{2}{{x}^{5}}\right)}\]

   if -107.46436521468802 < x < 11426.70565163402

   1. Initial program 0.1

      \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 1.5

      \[\leadsto \left(\frac{1}{x + 1} - \frac{2}{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}\right) + \frac{1}{x - 1}\]

   4. Applied add-cube-cbrt 2.3

      \[\leadsto \left(\frac{1}{x + 1} - \frac{\color{blue}{\left(\sqrt[3]{2} \cdot \sqrt[3]{2}\right) \cdot \sqrt[3]{2}}}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}\right) + \frac{1}{x - 1}\]

   5. Applied times-frac 2.4

      \[\leadsto \left(\frac{1}{x + 1} - \color{blue}{\frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{x}}}\right) + \frac{1}{x - 1}\]

   6. Applied add-sqr-sqrt 2.7

      \[\leadsto \left(\frac{1}{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1}}} - \frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right) + \frac{1}{x - 1}\]

   7. Applied *-un-lft-identity 2.7

      \[\leadsto \left(\frac{\color{blue}{1 \cdot 1}}{\sqrt{x + 1} \cdot \sqrt{x + 1}} - \frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right) + \frac{1}{x - 1}\]

   8. Applied times-frac 2.7

      \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{x + 1}} \cdot \frac{1}{\sqrt{x + 1}}} - \frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right) + \frac{1}{x - 1}\]

   9. Applied prod-diff 2.7

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\left(\frac{1}{\sqrt{x + 1}}\right), \left(\frac{1}{\sqrt{x + 1}}\right), \left(-\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right)\right) + \mathsf{fma}\left(\left(-\frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right), \left(\frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right), \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right)\right)\right)} + \frac{1}{x - 1}\]

   10. Applied associate-+l+ 2.7

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\frac{1}{\sqrt{x + 1}}\right), \left(\frac{1}{\sqrt{x + 1}}\right), \left(-\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right)\right) + \left(\mathsf{fma}\left(\left(-\frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right), \left(\frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right), \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right)\right) + \frac{1}{x - 1}\right)}\]

   11. Simplified 0.1

       \[\leadsto \color{blue}{\left(\frac{1}{x + 1} - \frac{2}{x}\right)} + \left(\mathsf{fma}\left(\left(-\frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right), \left(\frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right), \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right)\right) + \frac{1}{x - 1}\right)\]
   12. Using strategy rm

   13. Applied flip3-+ 37.7

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

   14. Applied frac-sub 37.7

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

   15. Applied frac-add 37.8

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

   16. Simplified 59.1

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(x - \mathsf{fma}\left(x, 2, 2\right)\right), \left(\mathsf{fma}\left(\left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right) - \frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right)\right), \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right) - \frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right)\right), \left(\frac{1}{x - 1} \cdot \left(\frac{1}{x - 1} - \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right) - \frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right)\right)\right)\right)\right)\right), \left(\mathsf{fma}\left(\left(\frac{\frac{1}{x - 1}}{x - 1}\right), \left(\frac{1}{x - 1}\right), \left(\left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right) - \frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right)\right) \cdot \left(\left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right) - \frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right)\right) \cdot \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right) - \frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right)\right)\right)\right)\right) \cdot \mathsf{fma}\left(x, x, x\right)\right)\right)}}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(\mathsf{fma}\left(\left(-\frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right), \left(\frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right), \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right)\right) \cdot \mathsf{fma}\left(\left(-\frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right), \left(\frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right), \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right)\right) + \left(\frac{1}{x - 1} \cdot \frac{1}{x - 1} - \mathsf{fma}\left(\left(-\frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right), \left(\frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right), \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right)\right) \cdot \frac{1}{x - 1}\right)\right)}\]

   17. Simplified 0.1

       \[\leadsto \frac{\mathsf{fma}\left(\left(x - \mathsf{fma}\left(x, 2, 2\right)\right), \left(\mathsf{fma}\left(\left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right) - \frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right)\right), \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right) - \frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right)\right), \left(\frac{1}{x - 1} \cdot \left(\frac{1}{x - 1} - \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right) - \frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right)\right)\right)\right)\right)\right), \left(\mathsf{fma}\left(\left(\frac{\frac{1}{x - 1}}{x - 1}\right), \left(\frac{1}{x - 1}\right), \left(\left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right) - \frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right)\right) \cdot \left(\left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right) - \frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right)\right) \cdot \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right) - \frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right)\right)\right)\right)\right) \cdot \mathsf{fma}\left(x, x, x\right)\right)\right)}{\color{blue}{\mathsf{fma}\left(\left(\left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right) \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{x}} - \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right) \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right), \left(\left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right) \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{x}} - \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right) \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right), \left(\left(\frac{1}{x - 1} - \left(\left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right) \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{x}} - \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right) \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right)\right) \cdot \frac{1}{x - 1}\right)\right) \cdot \mathsf{fma}\left(x, x, x\right)}}\]

   if 11426.70565163402 < x

   1. Initial program 20.2

      \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 50.8

      \[\leadsto \left(\frac{1}{x + 1} - \frac{2}{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}\right) + \frac{1}{x - 1}\]

   4. Applied add-cube-cbrt 55.1

      \[\leadsto \left(\frac{1}{x + 1} - \frac{\color{blue}{\left(\sqrt[3]{2} \cdot \sqrt[3]{2}\right) \cdot \sqrt[3]{2}}}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}\right) + \frac{1}{x - 1}\]

   5. Applied times-frac 56.2

      \[\leadsto \left(\frac{1}{x + 1} - \color{blue}{\frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{x}}}\right) + \frac{1}{x - 1}\]

   6. Applied add-sqr-sqrt 57.0

      \[\leadsto \left(\frac{1}{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1}}} - \frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right) + \frac{1}{x - 1}\]

   7. Applied *-un-lft-identity 57.0

      \[\leadsto \left(\frac{\color{blue}{1 \cdot 1}}{\sqrt{x + 1} \cdot \sqrt{x + 1}} - \frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right) + \frac{1}{x - 1}\]

   8. Applied times-frac 57.5

      \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{x + 1}} \cdot \frac{1}{\sqrt{x + 1}}} - \frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right) + \frac{1}{x - 1}\]

   9. Applied prod-diff 57.5

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\left(\frac{1}{\sqrt{x + 1}}\right), \left(\frac{1}{\sqrt{x + 1}}\right), \left(-\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right)\right) + \mathsf{fma}\left(\left(-\frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right), \left(\frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right), \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right)\right)\right)} + \frac{1}{x - 1}\]

   10. Applied associate-+l+ 57.5

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\frac{1}{\sqrt{x + 1}}\right), \left(\frac{1}{\sqrt{x + 1}}\right), \left(-\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right)\right) + \left(\mathsf{fma}\left(\left(-\frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right), \left(\frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right), \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right)\right) + \frac{1}{x - 1}\right)}\]

   11. Simplified 40.4

       \[\leadsto \color{blue}{\left(\frac{1}{x + 1} - \frac{2}{x}\right)} + \left(\mathsf{fma}\left(\left(-\frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right), \left(\frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right), \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right)\right) + \frac{1}{x - 1}\right)\]
   12. Using strategy rm

   13. Applied add-cube-cbrt 55.3

       \[\leadsto \left(\frac{1}{x + 1} - \color{blue}{\left(\sqrt[3]{\frac{2}{x}} \cdot \sqrt[3]{\frac{2}{x}}\right) \cdot \sqrt[3]{\frac{2}{x}}}\right) + \left(\mathsf{fma}\left(\left(-\frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right), \left(\frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right), \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right)\right) + \frac{1}{x - 1}\right)\]

   14. Applied *-un-lft-identity 55.3

       \[\leadsto \left(\frac{1}{\color{blue}{1 \cdot \left(x + 1\right)}} - \left(\sqrt[3]{\frac{2}{x}} \cdot \sqrt[3]{\frac{2}{x}}\right) \cdot \sqrt[3]{\frac{2}{x}}\right) + \left(\mathsf{fma}\left(\left(-\frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right), \left(\frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right), \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right)\right) + \frac{1}{x - 1}\right)\]

   15. Applied *-un-lft-identity 55.3

       \[\leadsto \left(\frac{\color{blue}{1 \cdot 1}}{1 \cdot \left(x + 1\right)} - \left(\sqrt[3]{\frac{2}{x}} \cdot \sqrt[3]{\frac{2}{x}}\right) \cdot \sqrt[3]{\frac{2}{x}}\right) + \left(\mathsf{fma}\left(\left(-\frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right), \left(\frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right), \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right)\right) + \frac{1}{x - 1}\right)\]

   16. Applied times-frac 55.3

       \[\leadsto \left(\color{blue}{\frac{1}{1} \cdot \frac{1}{x + 1}} - \left(\sqrt[3]{\frac{2}{x}} \cdot \sqrt[3]{\frac{2}{x}}\right) \cdot \sqrt[3]{\frac{2}{x}}\right) + \left(\mathsf{fma}\left(\left(-\frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right), \left(\frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right), \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right)\right) + \frac{1}{x - 1}\right)\]

   17. Applied prod-diff 55.2

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\left(\frac{1}{1}\right), \left(\frac{1}{x + 1}\right), \left(-\sqrt[3]{\frac{2}{x}} \cdot \left(\sqrt[3]{\frac{2}{x}} \cdot \sqrt[3]{\frac{2}{x}}\right)\right)\right) + \mathsf{fma}\left(\left(-\sqrt[3]{\frac{2}{x}}\right), \left(\sqrt[3]{\frac{2}{x}} \cdot \sqrt[3]{\frac{2}{x}}\right), \left(\sqrt[3]{\frac{2}{x}} \cdot \left(\sqrt[3]{\frac{2}{x}} \cdot \sqrt[3]{\frac{2}{x}}\right)\right)\right)\right)} + \left(\mathsf{fma}\left(\left(-\frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right), \left(\frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right), \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right)\right) + \frac{1}{x - 1}\right)\]

   18. Applied associate-+l+ 55.2

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\frac{1}{1}\right), \left(\frac{1}{x + 1}\right), \left(-\sqrt[3]{\frac{2}{x}} \cdot \left(\sqrt[3]{\frac{2}{x}} \cdot \sqrt[3]{\frac{2}{x}}\right)\right)\right) + \left(\mathsf{fma}\left(\left(-\sqrt[3]{\frac{2}{x}}\right), \left(\sqrt[3]{\frac{2}{x}} \cdot \sqrt[3]{\frac{2}{x}}\right), \left(\sqrt[3]{\frac{2}{x}} \cdot \left(\sqrt[3]{\frac{2}{x}} \cdot \sqrt[3]{\frac{2}{x}}\right)\right)\right) + \left(\mathsf{fma}\left(\left(-\frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right), \left(\frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right), \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right)\right) + \frac{1}{x - 1}\right)\right)}\]

   19. Simplified 44.0

       \[\leadsto \color{blue}{\left(\frac{1}{1 + x} - \frac{2}{x}\right)} + \left(\mathsf{fma}\left(\left(-\sqrt[3]{\frac{2}{x}}\right), \left(\sqrt[3]{\frac{2}{x}} \cdot \sqrt[3]{\frac{2}{x}}\right), \left(\sqrt[3]{\frac{2}{x}} \cdot \left(\sqrt[3]{\frac{2}{x}} \cdot \sqrt[3]{\frac{2}{x}}\right)\right)\right) + \left(\mathsf{fma}\left(\left(-\frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right), \left(\frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right), \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right)\right) + \frac{1}{x - 1}\right)\right)\]

   20. Taylor expanded around -inf 0.8

       \[\leadsto \color{blue}{2 \cdot \frac{1}{{x}^{3}}}\]

   21. Simplified 0.3

       \[\leadsto \color{blue}{\frac{\frac{2}{x \cdot x}}{x}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -107.46436521468802:\\ \;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{2}{{x}^{5}} + \frac{\frac{2}{x}}{x \cdot x}\right)\\ \mathbf{elif}\;x \le 11426.70565163402:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(x - \mathsf{fma}\left(x, 2, 2\right)\right), \left(\mathsf{fma}\left(\left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right) - \frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right)\right), \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right) - \frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right)\right), \left(\left(\frac{1}{x - 1} - \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right) - \frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right)\right)\right) \cdot \frac{1}{x - 1}\right)\right)\right), \left(\mathsf{fma}\left(x, x, x\right) \cdot \mathsf{fma}\left(\left(\frac{\frac{1}{x - 1}}{x - 1}\right), \left(\frac{1}{x - 1}\right), \left(\left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right) - \frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right)\right) \cdot \left(\left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right) - \frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right)\right) \cdot \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right) - \frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right)\right)\right)\right)\right)\right)\right)}{\mathsf{fma}\left(\left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right) - \frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right)\right), \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right) - \frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right)\right), \left(\left(\frac{1}{x - 1} - \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right) - \frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \left(\frac{\sqrt[3]{2}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{x}}\right)\right)\right) \cdot \frac{1}{x - 1}\right)\right) \cdot \mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{x \cdot x}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019124 +o rules:numerics
(FPCore (x)
  :name "3frac (problem 3.3.3)"

  :herbie-target
  (/ 2 (* x (- (* x x) 1)))

  (+ (- (/ 1 (+ x 1)) (/ 2 x)) (/ 1 (- x 1))))