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Derivation

1. Initial program 29.3

   \[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
2. Using strategy rm

3. Applied flip3--29.3

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

5. Applied add-cube-cbrt29.5

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

6. Applied unpow-prod-down29.5

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

7. Applied add-cube-cbrt29.5

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

8. Applied cbrt-prod29.5

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

9. Applied unpow-prod-down29.5

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

10. Applied prod-diff29.5

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

11. Simplified29.4

    \[\leadsto \frac{\color{blue}{\left(1 + 0\right)} + (\left(-{\left(\sqrt[3]{\sqrt[3]{x}}\right)}^{3}\right) \cdot \left({\left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right)}^{3}\right) + \left({\left(\sqrt[3]{\sqrt[3]{x}}\right)}^{3} \cdot {\left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right)}^{3}\right))_*}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)}\]

12. Simplified0.5

    \[\leadsto \frac{\left(1 + 0\right) + \color{blue}{0}}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)}\]

13. Final simplification0.5

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

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed 2018216 +o rules:numerics
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))