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Derivation

1. Split input into 2 regimes

2. if x < -0.9924714463196989 or 4122.6992300157335 < x

   1. Initial program 59.8

      \[\sqrt[3]{x + 1} - \sqrt[3]{x}\]

   2. Initial simplification59.8

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

   3. Taylor expanded around -inf 62.4

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

   4. Simplified0.9

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

   if -0.9924714463196989 < x < 4122.6992300157335

   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

      \[\leadsto \color{blue}{\sqrt{\sqrt[3]{1 + x}} \cdot \sqrt{\sqrt[3]{1 + x}}} - \sqrt[3]{x}\]
3. Recombined 2 regimes into one program.

4. Final simplification0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.9924714463196989 \lor \neg \left(x \le 4122.6992300157335\right):\\ \;\;\;\;\frac{\sqrt[3]{x}}{x} \cdot \left(\left(\frac{1}{3} + \frac{\frac{-1}{9}}{x}\right) + \frac{\frac{\frac{5}{81}}{x}}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\sqrt[3]{1 + x}} \cdot \sqrt{\sqrt[3]{1 + x}} - \sqrt[3]{x}\\ \end{array}\]

Runtime

Time bar (total: 17.9s)Debug logProfile

herbie shell --seed 2018252 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))