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Derivation

1. Split input into 2 regimes

2. if x < 5674.445687910233

   1. Initial program 0.1

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

   3. Applied pow1/30.1

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

   5. Applied flip3--0.2

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

   if 5674.445687910233 < x

   1. Initial program 60.2

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

   3. Applied pow1/359.6

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

   4. Taylor expanded around -inf 62.4

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

   5. Simplified0.6

      \[\leadsto \color{blue}{\left(\left(\frac{1}{3} + \frac{\frac{-1}{9}}{x}\right) + \frac{\frac{\frac{5}{81}}{x}}{x}\right) \cdot \frac{\sqrt[3]{x}}{x}}\]
   6. Using strategy rm

   7. Applied div-inv0.7

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

   8. Applied associate-*r*0.7

      \[\leadsto \color{blue}{\left(\left(\left(\frac{1}{3} + \frac{\frac{-1}{9}}{x}\right) + \frac{\frac{\frac{5}{81}}{x}}{x}\right) \cdot \sqrt[3]{x}\right) \cdot \frac{1}{x}}\]
   9. Using strategy rm

   10. Applied add-cbrt-cube0.7

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

   11. Applied cbrt-unprod0.6

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

4. Final simplification0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 5674.445687910233:\\ \;\;\;\;\frac{{\left(\sqrt[3]{1 + x}\right)}^{3} - {\left({x}^{\frac{1}{3}}\right)}^{3}}{\left({x}^{\frac{1}{3}} \cdot {x}^{\frac{1}{3}} + {x}^{\frac{1}{3}} \cdot \sqrt[3]{1 + x}\right) + \sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \sqrt[3]{x \cdot \left(\left(\left(\frac{\frac{-1}{9}}{x} + \frac{1}{3}\right) + \frac{\frac{\frac{5}{81}}{x}}{x}\right) \cdot \left(\left(\left(\frac{\frac{-1}{9}}{x} + \frac{1}{3}\right) + \frac{\frac{\frac{5}{81}}{x}}{x}\right) \cdot \left(\left(\frac{\frac{-1}{9}}{x} + \frac{1}{3}\right) + \frac{\frac{\frac{5}{81}}{x}}{x}\right)\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019007 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))

Details

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