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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -0.9894432219097609:\\ \;\;\;\;\frac{(\left(\sqrt[3]{\frac{\frac{\frac{1}{x}}{x}}{x \cdot x}}\right) \cdot \frac{-1}{9} + \left((\frac{2}{3} \cdot \left(\sqrt[3]{\frac{1}{x}}\right) + \left(\sqrt[3]{\frac{1}{{x}^{7}}} \cdot \frac{4}{81}\right))_*\right))_*}{\sqrt[3]{x} + \sqrt[3]{1 + x}}\\ \mathbf{elif}\;x \le 4049.184576509551:\\ \;\;\;\;(\left(\sqrt{\frac{\sqrt[3]{(x \cdot x + -1)_*}}{\sqrt[3]{x - 1}}}\right) \cdot \left(\sqrt{\sqrt[3]{1 + x}}\right) + \left(-\sqrt[3]{x}\right))_*\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left(\sqrt[3]{\frac{\frac{\frac{1}{x}}{x}}{x \cdot x}}\right) \cdot \frac{-1}{9} + \left((\frac{2}{3} \cdot \left(\sqrt[3]{\frac{1}{x}}\right) + \left(\sqrt[3]{\frac{1}{{x}^{7}}} \cdot \frac{4}{81}\right))_*\right))_*}{\sqrt[3]{x} + \sqrt[3]{1 + x}}\\ \end{array}\]

Derivation

Split input into 2 regimes
if x < -0.9894432219097609 or 4049.184576509551 < x
1. Initial program 59.9
  \[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
2. Using strategy rm
3. Applied flip--59.9
  \[\leadsto \color{blue}{\frac{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} - \sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{x + 1} + \sqrt[3]{x}}}\]
4. Taylor expanded around inf 33.8
  \[\leadsto \frac{\color{blue}{\left(\frac{4}{81} \cdot {\left(\frac{1}{{x}^{7}}\right)}^{\frac{1}{3}} + \frac{2}{3} \cdot {\left(\frac{1}{x}\right)}^{\frac{1}{3}}\right) - \frac{1}{9} \cdot {\left(\frac{1}{{x}^{4}}\right)}^{\frac{1}{3}}}}{\sqrt[3]{x + 1} + \sqrt[3]{x}}\]
5. Simplified1.1
  \[\leadsto \frac{\color{blue}{(\left(\sqrt[3]{\frac{\frac{\frac{1}{x}}{x}}{x \cdot x}}\right) \cdot \frac{-1}{9} + \left((\frac{2}{3} \cdot \left(\sqrt[3]{\frac{1}{x}}\right) + \left(\sqrt[3]{\frac{1}{{x}^{7}}} \cdot \frac{4}{81}\right))_*\right))_*}}{\sqrt[3]{x + 1} + \sqrt[3]{x}}\]
if -0.9894432219097609 < x < 4049.184576509551
1. Initial program 0.1
  \[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
2. Using strategy rm
3. Applied add-sqr-sqrt0.1
  \[\leadsto \color{blue}{\sqrt{\sqrt[3]{x + 1}} \cdot \sqrt{\sqrt[3]{x + 1}}} - \sqrt[3]{x}\]
4. Applied fma-neg0.1
  \[\leadsto \color{blue}{(\left(\sqrt{\sqrt[3]{x + 1}}\right) \cdot \left(\sqrt{\sqrt[3]{x + 1}}\right) + \left(-\sqrt[3]{x}\right))_*}\]
5. Using strategy rm
6. Applied flip-+0.1
  \[\leadsto (\left(\sqrt{\sqrt[3]{\color{blue}{\frac{x \cdot x - 1 \cdot 1}{x - 1}}}}\right) \cdot \left(\sqrt{\sqrt[3]{x + 1}}\right) + \left(-\sqrt[3]{x}\right))_*\]
7. Applied cbrt-div0.1
  \[\leadsto (\left(\sqrt{\color{blue}{\frac{\sqrt[3]{x \cdot x - 1 \cdot 1}}{\sqrt[3]{x - 1}}}}\right) \cdot \left(\sqrt{\sqrt[3]{x + 1}}\right) + \left(-\sqrt[3]{x}\right))_*\]
8. Simplified0.1
  \[\leadsto (\left(\sqrt{\frac{\color{blue}{\sqrt[3]{(x \cdot x + -1)_*}}}{\sqrt[3]{x - 1}}}\right) \cdot \left(\sqrt{\sqrt[3]{x + 1}}\right) + \left(-\sqrt[3]{x}\right))_*\]
Recombined 2 regimes into one program.
Final simplification0.6
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.9894432219097609:\\ \;\;\;\;\frac{(\left(\sqrt[3]{\frac{\frac{\frac{1}{x}}{x}}{x \cdot x}}\right) \cdot \frac{-1}{9} + \left((\frac{2}{3} \cdot \left(\sqrt[3]{\frac{1}{x}}\right) + \left(\sqrt[3]{\frac{1}{{x}^{7}}} \cdot \frac{4}{81}\right))_*\right))_*}{\sqrt[3]{x} + \sqrt[3]{1 + x}}\\ \mathbf{elif}\;x \le 4049.184576509551:\\ \;\;\;\;(\left(\sqrt{\frac{\sqrt[3]{(x \cdot x + -1)_*}}{\sqrt[3]{x - 1}}}\right) \cdot \left(\sqrt{\sqrt[3]{1 + x}}\right) + \left(-\sqrt[3]{x}\right))_*\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left(\sqrt[3]{\frac{\frac{\frac{1}{x}}{x}}{x \cdot x}}\right) \cdot \frac{-1}{9} + \left((\frac{2}{3} \cdot \left(\sqrt[3]{\frac{1}{x}}\right) + \left(\sqrt[3]{\frac{1}{{x}^{7}}} \cdot \frac{4}{81}\right))_*\right))_*}{\sqrt[3]{x} + \sqrt[3]{1 + x}}\\ \end{array}\]

Error

Derivation

`if x < -0.9894432219097609 or 4049.184576509551 < x`

`if -0.9894432219097609 < x < 4049.184576509551`

Reproduce