Derivation

Split input into 2 regimes
if x < 5338.0242829215695
1. Initial program 0.1
  \[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
2. Initial simplification0.1
  \[\leadsto \sqrt[3]{1 + x} - \sqrt[3]{x}\]
3. Using strategy rm
4. Applied add-sqr-sqrt0.1
  \[\leadsto \sqrt[3]{1 + x} - \color{blue}{\sqrt{\sqrt[3]{x}} \cdot \sqrt{\sqrt[3]{x}}}\]
if 5338.0242829215695 < x
1. Initial program 60.1
  \[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
2. Initial simplification60.1
  \[\leadsto \sqrt[3]{1 + x} - \sqrt[3]{x}\]
3. Taylor expanded around inf 33.5
  \[\leadsto \color{blue}{\left(\frac{1}{3} \cdot {\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}} + \frac{5}{81} \cdot {\left(\frac{1}{{x}^{8}}\right)}^{\frac{1}{3}}\right) - \frac{1}{9} \cdot {\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}}\]
4. Simplified31.4
  \[\leadsto \color{blue}{(\left(\sqrt[3]{\frac{1}{{x}^{5}}}\right) \cdot \frac{-1}{9} + \left((\frac{1}{3} \cdot \left(\sqrt[3]{\frac{1}{x \cdot x}}\right) + \left(\sqrt[3]{\frac{1}{{x}^{8}}} \cdot \frac{5}{81}\right))_*\right))_*}\]
Recombined 2 regimes into one program.
Final simplification15.9
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le 5338.0242829215695:\\ \;\;\;\;\sqrt[3]{1 + x} - \sqrt{\sqrt[3]{x}} \cdot \sqrt{\sqrt[3]{x}}\\ \mathbf{else}:\\ \;\;\;\;(\left(\sqrt[3]{\frac{1}{{x}^{5}}}\right) \cdot \frac{-1}{9} + \left((\frac{1}{3} \cdot \left(\sqrt[3]{\frac{1}{x \cdot x}}\right) + \left(\frac{5}{81} \cdot \sqrt[3]{\frac{1}{{x}^{8}}}\right))_*\right))_*\\ \end{array}\]

Runtime

herbie shell --seed 2018346 +o rules:numerics
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))