Error

Derivation

1. Split input into 2 regimes

2. if (- (cbrt (+ x 1)) (cbrt x)) < 4.026806576007402e-05

   1. Initial program 60.5

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

   2. Taylor expanded around -inf 62.4

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

   3. Simplified0.6

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

   if 4.026806576007402e-05 < (- (cbrt (+ x 1)) (cbrt x))

   1. Initial program 0.2

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

   3. Applied add-cube-cbrt0.3

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

4. Final simplification0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{1 + x} - \sqrt[3]{x} \le 4.026806576007402 \cdot 10^{-05}:\\ \;\;\;\;\left(\sqrt[3]{x} - \sqrt[3]{-1} \cdot \sqrt[3]{-x}\right) + \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right) \cdot \frac{\sqrt[3]{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\sqrt[3]{1 + x}} \cdot \left(\sqrt[3]{\sqrt[3]{1 + x}} \cdot \sqrt[3]{\sqrt[3]{1 + x}}\right) - \sqrt[3]{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019093 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))