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Derivation

1. Initial program 29.6

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

3. Applied add-cbrt-cube29.7

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

5. Applied flip3--29.6

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

6. Taylor expanded around inf 0.6

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

7. Final simplification0.6

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

Runtime

Time bar (total: 45.9s)Debug logProfile

herbie shell --seed 2018274 +o rules:numerics
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))