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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -7070.912443794031:\\ \;\;\;\;\frac{(\left(\sqrt[3]{\frac{\frac{\frac{1}{x}}{x}}{x \cdot x}}\right) \cdot \frac{-1}{9} + \left((\frac{2}{3} \cdot \left(\sqrt[3]{\frac{1}{x}}\right) + \left(\sqrt[3]{\frac{1}{{x}^{7}}} \cdot \frac{4}{81}\right))_*\right))_*}{\sqrt[3]{x} + \sqrt[3]{1 + x}}\\ \mathbf{elif}\;x \le 2293.0539734593835:\\ \;\;\;\;(\left(\sqrt[3]{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}}\right) \cdot \left(\log_* (1 + (e^{\sqrt[3]{\sqrt[3]{1 + x}}} - 1)^*)\right) + \left(-\sqrt[3]{x}\right))_*\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left(\sqrt[3]{\frac{\frac{\frac{1}{x}}{x}}{x \cdot x}}\right) \cdot \frac{-1}{9} + \left((\frac{2}{3} \cdot \left(\sqrt[3]{\frac{1}{x}}\right) + \left(\sqrt[3]{\frac{1}{{x}^{7}}} \cdot \frac{4}{81}\right))_*\right))_*}{\sqrt[3]{x} + \sqrt[3]{1 + x}}\\ \end{array}\]

Derivation

Split input into 2 regimes
if x < -7070.912443794031 or 2293.0539734593835 < x
1. Initial program 60.0
  \[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
2. Using strategy rm
3. Applied flip--60.0
  \[\leadsto \color{blue}{\frac{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} - \sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{x + 1} + \sqrt[3]{x}}}\]
4. Taylor expanded around 0 60.8
  \[\leadsto \frac{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} - \color{blue}{{x}^{\frac{1}{3}}} \cdot \sqrt[3]{x}}{\sqrt[3]{x + 1} + \sqrt[3]{x}}\]
5. Simplified60.0
  \[\leadsto \frac{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} - \color{blue}{\sqrt[3]{x}} \cdot \sqrt[3]{x}}{\sqrt[3]{x + 1} + \sqrt[3]{x}}\]
6. Taylor expanded around inf 33.8
  \[\leadsto \frac{\color{blue}{\left(\frac{4}{81} \cdot {\left(\frac{1}{{x}^{7}}\right)}^{\frac{1}{3}} + \frac{2}{3} \cdot {\left(\frac{1}{x}\right)}^{\frac{1}{3}}\right) - \frac{1}{9} \cdot {\left(\frac{1}{{x}^{4}}\right)}^{\frac{1}{3}}}}{\sqrt[3]{x + 1} + \sqrt[3]{x}}\]
7. Simplified1.0
  \[\leadsto \frac{\color{blue}{(\left(\sqrt[3]{\frac{\frac{\frac{1}{x}}{x}}{x \cdot x}}\right) \cdot \frac{-1}{9} + \left((\frac{2}{3} \cdot \left(\sqrt[3]{\frac{1}{x}}\right) + \left(\sqrt[3]{\frac{1}{{x}^{7}}} \cdot \frac{4}{81}\right))_*\right))_*}}{\sqrt[3]{x + 1} + \sqrt[3]{x}}\]
if -7070.912443794031 < x < 2293.0539734593835
1. Initial program 0.1
  \[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
2. Using strategy rm
3. Applied add-cube-cbrt0.1
  \[\leadsto \sqrt[3]{\color{blue}{\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right) \cdot \sqrt[3]{x + 1}}} - \sqrt[3]{x}\]
4. Applied cbrt-prod0.1
  \[\leadsto \color{blue}{\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}}} - \sqrt[3]{x}\]
5. Applied fma-neg0.1
  \[\leadsto \color{blue}{(\left(\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{x + 1}}\right) + \left(-\sqrt[3]{x}\right))_*}\]
6. Using strategy rm
7. Applied log1p-expm1-u0.1
  \[\leadsto (\left(\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}\right) \cdot \color{blue}{\left(\log_* (1 + (e^{\sqrt[3]{\sqrt[3]{x + 1}}} - 1)^*)\right)} + \left(-\sqrt[3]{x}\right))_*\]
Recombined 2 regimes into one program.
Final simplification0.6
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -7070.912443794031:\\ \;\;\;\;\frac{(\left(\sqrt[3]{\frac{\frac{\frac{1}{x}}{x}}{x \cdot x}}\right) \cdot \frac{-1}{9} + \left((\frac{2}{3} \cdot \left(\sqrt[3]{\frac{1}{x}}\right) + \left(\sqrt[3]{\frac{1}{{x}^{7}}} \cdot \frac{4}{81}\right))_*\right))_*}{\sqrt[3]{x} + \sqrt[3]{1 + x}}\\ \mathbf{elif}\;x \le 2293.0539734593835:\\ \;\;\;\;(\left(\sqrt[3]{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}}\right) \cdot \left(\log_* (1 + (e^{\sqrt[3]{\sqrt[3]{1 + x}}} - 1)^*)\right) + \left(-\sqrt[3]{x}\right))_*\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left(\sqrt[3]{\frac{\frac{\frac{1}{x}}{x}}{x \cdot x}}\right) \cdot \frac{-1}{9} + \left((\frac{2}{3} \cdot \left(\sqrt[3]{\frac{1}{x}}\right) + \left(\sqrt[3]{\frac{1}{{x}^{7}}} \cdot \frac{4}{81}\right))_*\right))_*}{\sqrt[3]{x} + \sqrt[3]{1 + x}}\\ \end{array}\]

Error

Derivation

`if x < -7070.912443794031 or 2293.0539734593835 < x`

`if -7070.912443794031 < x < 2293.0539734593835`

Reproduce