Error

Derivation

1. Split input into 2 regimes

2. if x < -7506.02202390481 or 2668.8620780384454 < x

   1. Initial program 60.2

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

   3. Applied flip--60.2

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

   4. Taylor expanded around 0 60.9

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

   5. Simplified60.2

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

   6. Taylor expanded around inf 33.4

      \[\leadsto \frac{\color{blue}{\left(\frac{4}{81} \cdot {\left(\frac{1}{{x}^{7}}\right)}^{\frac{1}{3}} + \frac{2}{3} \cdot {\left(\frac{1}{x}\right)}^{\frac{1}{3}}\right) - \frac{1}{9} \cdot {\left(\frac{1}{{x}^{4}}\right)}^{\frac{1}{3}}}}{\sqrt[3]{x + 1} + \sqrt[3]{x}}\]

   7. Simplified1.0

      \[\leadsto \frac{\color{blue}{(\frac{-1}{9} \cdot \left(\sqrt[3]{\frac{1}{{x}^{4}}}\right) + \left((\left(\sqrt[3]{\frac{1}{{x}^{7}}}\right) \cdot \frac{4}{81} + \left(\sqrt[3]{\frac{1}{x}} \cdot \frac{2}{3}\right))_*\right))_*}}{\sqrt[3]{x + 1} + \sqrt[3]{x}}\]

   if -7506.02202390481 < x < 2668.8620780384454

   1. Initial program 0.1

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

   3. Applied flip-+0.1

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

   4. Applied cbrt-div0.1

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

   5. Simplified0.1

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

4. Final simplification0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -7506.02202390481 \lor \neg \left(x \le 2668.8620780384454\right):\\ \;\;\;\;\frac{(\frac{-1}{9} \cdot \left(\sqrt[3]{\frac{1}{{x}^{4}}}\right) + \left((\left(\sqrt[3]{\frac{1}{{x}^{7}}}\right) \cdot \frac{4}{81} + \left(\sqrt[3]{\frac{1}{x}} \cdot \frac{2}{3}\right))_*\right))_*}{\sqrt[3]{x} + \sqrt[3]{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{(x \cdot x + -1)_*}}{\sqrt[3]{x - 1}} - \sqrt[3]{x}\\ \end{array}\]

Runtime

Time bar (total: 16.6s)Debug logProfile

herbie shell --seed 2018286 +o rules:numerics
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))