Error

Try it out

Derivation

1. Split input into 2 regimes

2. if x < 4072.6667673670877

   1. Initial program 0.1

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

   3. Applied add-cube-cbrt0.2

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

   if 4072.6667673670877 < x

   1. Initial program 60.2

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

   3. Applied add-cbrt-cube60.1

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

   5. Applied add-cube-cbrt60.1

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

   7. Applied add-sqr-sqrt60.1

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

   8. Applied cbrt-prod60.1

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

   9. Taylor expanded around inf 33.6

      \[\leadsto \color{blue}{\left(\frac{1}{3} \cdot {\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}} + \frac{5}{81} \cdot {\left(\frac{1}{{x}^{8}}\right)}^{\frac{1}{3}}\right) - \frac{1}{9} \cdot {\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}}\]

   10. Simplified31.5

       \[\leadsto \color{blue}{\left(\frac{1}{3} \cdot \sqrt[3]{\frac{1}{x \cdot x}} - \frac{1}{9} \cdot \sqrt[3]{\frac{1}{{x}^{5}}}\right) + \sqrt[3]{\frac{1}{{x}^{8}}} \cdot \frac{5}{81}}\]
3. Recombined 2 regimes into one program.

4. Final simplification15.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 4072.6667673670877:\\ \;\;\;\;\sqrt[3]{1 + x} - \sqrt[3]{\sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{5}{81} \cdot \sqrt[3]{\frac{1}{{x}^{8}}} + \left(\sqrt[3]{\frac{1}{x \cdot x}} \cdot \frac{1}{3} - \frac{1}{9} \cdot \sqrt[3]{\frac{1}{{x}^{5}}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019008 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))

Details

Time bar (total: 2.5m)Debug log