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Derivation

1. Initial program 29.0

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

3. Applied flip3--28.9

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

4. Taylor expanded around -inf 0.5

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

6. Applied add-sqr-sqrt0.6

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

7. Applied associate-*r*0.5

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

8. Final simplification0.5

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

Reproduce

herbie shell --seed 2018365 +o rules:numerics
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))

Details

Time bar (total: 47.8s)Debug log