Average Error: 9.7 → 0.2
Time: 3.1m
Precision: 64
Internal Precision: 128
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -137.20814296538708:\\ \;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{2}{x} \cdot \frac{\frac{1}{x}}{x} + \frac{2}{{x}^{5}}\right)\\ \mathbf{elif}\;x \le 112.50341112112116:\\ \;\;\;\;\frac{(\left(x - (x \cdot 2 + 2)_*\right) \cdot \left((\left(\frac{1}{x - 1} - \left(\frac{\frac{2}{\sqrt[3]{x}}}{\sqrt[3]{x} \cdot \sqrt[3]{x}} - \frac{\frac{2}{\sqrt[3]{x}}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right)\right) \cdot \left(\frac{1}{x - 1}\right) + \left(\left(\frac{\frac{2}{\sqrt[3]{x}}}{\sqrt[3]{x} \cdot \sqrt[3]{x}} - \frac{\frac{2}{\sqrt[3]{x}}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right) \cdot \left(\frac{\frac{2}{\sqrt[3]{x}}}{\sqrt[3]{x} \cdot \sqrt[3]{x}} - \frac{\frac{2}{\sqrt[3]{x}}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right)\right))_*\right) + \left((x \cdot x + x)_* \cdot (\left(\frac{1}{x - 1} \cdot \frac{1}{x - 1}\right) \cdot \left(\frac{1}{x - 1}\right) + \left(\left(\left(\frac{\frac{2}{\sqrt[3]{x}}}{\sqrt[3]{x} \cdot \sqrt[3]{x}} - \frac{\frac{2}{\sqrt[3]{x}}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right) \cdot \left(\frac{\frac{2}{\sqrt[3]{x}}}{\sqrt[3]{x} \cdot \sqrt[3]{x}} - \frac{\frac{2}{\sqrt[3]{x}}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right)\right) \cdot \left(\frac{\frac{2}{\sqrt[3]{x}}}{\sqrt[3]{x} \cdot \sqrt[3]{x}} - \frac{\frac{2}{\sqrt[3]{x}}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right)\right))_*\right))_*}{(\left(\frac{1}{x - 1}\right) \cdot \left(\frac{1}{x - 1} - \left(\frac{2}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}} - \frac{2}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}\right)\right) + \left(\left(\frac{2}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}} - \frac{2}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}\right) \cdot \left(\frac{2}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}} - \frac{2}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}\right)\right))_* \cdot (x \cdot x + x)_*}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{2}{{x}^{5}} + \frac{2}{x \cdot \left(x \cdot x\right)}\right)\\ \end{array}\]

Error

Bits error versus x

Target

Original9.7
Target0.3
Herbie0.2
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -137.20814296538708

    1. Initial program 19.9

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

    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. Simplified0.1

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

    5. Applied div-inv0.1

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

    6. Applied times-frac0.2

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

    if -137.20814296538708 < x < 112.50341112112116

    1. Initial program 0.0

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

    3. Applied add-cube-cbrt1.4

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

    4. Applied *-un-lft-identity1.4

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

    5. Applied times-frac1.5

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

    6. Applied add-sqr-sqrt1.6

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

    7. Applied *-un-lft-identity1.6

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

    8. Applied times-frac1.6

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

    9. Applied prod-diff1.6

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

    10. Applied associate-+l+1.6

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

    11. Simplified0.0

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

    13. Applied flip3-+38.2

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

    14. Applied frac-sub38.2

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

    15. Applied frac-add38.3

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

    16. Simplified59.5

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

    17. Simplified0.0

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

    if 112.50341112112116 < x

    1. Initial program 20.4

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

    2. Taylor expanded around inf 0.6

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

    3. Simplified0.1

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

    4. Taylor expanded around inf 0.6

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

    5. Simplified0.6

      \[\leadsto \frac{2}{{x}^{7}} + \left(\color{blue}{\frac{2}{\left(x \cdot x\right) \cdot x}} + \frac{2}{{x}^{5}}\right)\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -137.20814296538708:\\ \;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{2}{x} \cdot \frac{\frac{1}{x}}{x} + \frac{2}{{x}^{5}}\right)\\ \mathbf{elif}\;x \le 112.50341112112116:\\ \;\;\;\;\frac{(\left(x - (x \cdot 2 + 2)_*\right) \cdot \left((\left(\frac{1}{x - 1} - \left(\frac{\frac{2}{\sqrt[3]{x}}}{\sqrt[3]{x} \cdot \sqrt[3]{x}} - \frac{\frac{2}{\sqrt[3]{x}}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right)\right) \cdot \left(\frac{1}{x - 1}\right) + \left(\left(\frac{\frac{2}{\sqrt[3]{x}}}{\sqrt[3]{x} \cdot \sqrt[3]{x}} - \frac{\frac{2}{\sqrt[3]{x}}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right) \cdot \left(\frac{\frac{2}{\sqrt[3]{x}}}{\sqrt[3]{x} \cdot \sqrt[3]{x}} - \frac{\frac{2}{\sqrt[3]{x}}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right)\right))_*\right) + \left((x \cdot x + x)_* \cdot (\left(\frac{1}{x - 1} \cdot \frac{1}{x - 1}\right) \cdot \left(\frac{1}{x - 1}\right) + \left(\left(\left(\frac{\frac{2}{\sqrt[3]{x}}}{\sqrt[3]{x} \cdot \sqrt[3]{x}} - \frac{\frac{2}{\sqrt[3]{x}}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right) \cdot \left(\frac{\frac{2}{\sqrt[3]{x}}}{\sqrt[3]{x} \cdot \sqrt[3]{x}} - \frac{\frac{2}{\sqrt[3]{x}}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right)\right) \cdot \left(\frac{\frac{2}{\sqrt[3]{x}}}{\sqrt[3]{x} \cdot \sqrt[3]{x}} - \frac{\frac{2}{\sqrt[3]{x}}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right)\right))_*\right))_*}{(\left(\frac{1}{x - 1}\right) \cdot \left(\frac{1}{x - 1} - \left(\frac{2}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}} - \frac{2}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}\right)\right) + \left(\left(\frac{2}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}} - \frac{2}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}\right) \cdot \left(\frac{2}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}} - \frac{2}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}\right)\right))_* \cdot (x \cdot x + x)_*}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{2}{{x}^{5}} + \frac{2}{x \cdot \left(x \cdot x\right)}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019094 +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))))