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Derivation

1. Split input into 3 regimes

2. if x < 3998.3190634746425

   1. Initial program 0.1

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

   3. Applied add-sqr-sqrt0.2

      \[\leadsto \color{blue}{\sqrt{\sqrt[3]{x + 1}} \cdot \sqrt{\sqrt[3]{x + 1}}} - \sqrt[3]{x}\]

   if 3998.3190634746425 < x < 1.3592927885672945e+154

   1. Initial program 59.2

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

   3. Applied add-cbrt-cube58.9

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

   4. Taylor expanded around inf 5.1

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

   5. Simplified0.9

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

   if 1.3592927885672945e+154 < x

   1. Initial program 61.0

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

   3. Applied add-cbrt-cube61.2

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

   5. Applied add-cube-cbrt61.4

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

   7. Applied add-exp-log61.4

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

   8. Taylor expanded around inf 6.9

      \[\leadsto e^{\color{blue}{\left(\log \left(\frac{1}{3} \cdot {x}^{\frac{1}{3}}\right) + \left(\log \left(\frac{1}{x}\right) + \frac{7}{54} \cdot \frac{1}{{x}^{2}}\right)\right) - \frac{1}{3} \cdot \frac{1}{x}}}\]

   9. Simplified7.6

      \[\leadsto e^{\color{blue}{\left(\log x \cdot \frac{1}{3} - \frac{\frac{1}{3}}{x}\right) + \left(\left(\log \frac{1}{3} - \log x\right) + \frac{\frac{7}{54}}{x \cdot x}\right)}}\]
3. Recombined 3 regimes into one program.

4. Final simplification2.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 3998.3190634746425:\\ \;\;\;\;\sqrt{\sqrt[3]{1 + x}} \cdot \sqrt{\sqrt[3]{1 + x}} - \sqrt[3]{x}\\ \mathbf{elif}\;x \le 1.3592927885672945 \cdot 10^{+154}:\\ \;\;\;\;\sqrt[3]{\frac{1}{{x}^{8}}} \cdot \frac{5}{81} + \left(\sqrt[3]{\frac{1}{x \cdot x}} \cdot \frac{1}{3} - \frac{1}{9} \cdot \sqrt[3]{\frac{1}{{x}^{5}}}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\frac{1}{3} \cdot \log x - \frac{\frac{1}{3}}{x}\right) + \left(\left(\log \frac{1}{3} - \log x\right) + \frac{\frac{7}{54}}{x \cdot x}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 1137770271 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))

Details

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