\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]

↓

\[\begin{array}{l} \mathbf{if}\;n \le -32555399.34096443:\\ \;\;\;\;(\left(\frac{\frac{-1}{2}}{x}\right) \cdot \left(\frac{1}{x \cdot n}\right) + \left(\frac{\frac{1}{n}}{x} - \frac{-\log x}{\left(x \cdot n\right) \cdot n}\right))_*\\ \mathbf{elif}\;n \le -1.61007232677794 \cdot 10^{-310}:\\ \;\;\;\;\left(\sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\\ \mathbf{elif}\;n \le 159200.81643882472:\\ \;\;\;\;e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{\frac{-1}{2}}{x}\right) \cdot \left(\frac{1}{x \cdot n}\right) + \left(\frac{\frac{1}{n}}{x} - \frac{-\log x}{\left(x \cdot n\right) \cdot n}\right))_*\\ \end{array}\]

Derivation

Split input into 3 regimes
if n < -32555399.34096443 or 159200.81643882472 < n
1. Initial program 44.7
  \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
2. Taylor expanded around inf 32.8
  \[\leadsto \color{blue}{\frac{1}{x \cdot n} - \left(\frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}} + \frac{1}{2} \cdot \frac{1}{{x}^{2} \cdot n}\right)}\]
3. Simplified32.7
  \[\leadsto \color{blue}{(\left(\frac{\frac{-1}{2}}{x}\right) \cdot \left(\frac{1}{n \cdot x}\right) + \left(\frac{1}{n \cdot x} - \frac{-\log x}{n \cdot \left(n \cdot x\right)}\right))_*}\]
4. Using strategy rm
5. Applied associate-/r*32.1
  \[\leadsto (\left(\frac{\frac{-1}{2}}{x}\right) \cdot \left(\frac{1}{n \cdot x}\right) + \left(\color{blue}{\frac{\frac{1}{n}}{x}} - \frac{-\log x}{n \cdot \left(n \cdot x\right)}\right))_*\]
if -32555399.34096443 < n < -1.61007232677794e-310
1. Initial program 0.7
  \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
2. Using strategy rm
3. Applied add-cube-cbrt0.7
  \[\leadsto \color{blue}{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\]
if -1.61007232677794e-310 < n < 159200.81643882472
1. Initial program 25.1
  \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
2. Using strategy rm
3. Applied add-exp-log25.1
  \[\leadsto {\color{blue}{\left(e^{\log \left(x + 1\right)}\right)}}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
4. Applied pow-exp25.1
  \[\leadsto \color{blue}{e^{\log \left(x + 1\right) \cdot \frac{1}{n}}} - {x}^{\left(\frac{1}{n}\right)}\]
5. Simplified1.4
  \[\leadsto e^{\color{blue}{\frac{\log_* (1 + x)}{n}}} - {x}^{\left(\frac{1}{n}\right)}\]
Recombined 3 regimes into one program.
Final simplification19.0
\[\leadsto \begin{array}{l} \mathbf{if}\;n \le -32555399.34096443:\\ \;\;\;\;(\left(\frac{\frac{-1}{2}}{x}\right) \cdot \left(\frac{1}{x \cdot n}\right) + \left(\frac{\frac{1}{n}}{x} - \frac{-\log x}{\left(x \cdot n\right) \cdot n}\right))_*\\ \mathbf{elif}\;n \le -1.61007232677794 \cdot 10^{-310}:\\ \;\;\;\;\left(\sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\\ \mathbf{elif}\;n \le 159200.81643882472:\\ \;\;\;\;e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{\frac{-1}{2}}{x}\right) \cdot \left(\frac{1}{x \cdot n}\right) + \left(\frac{\frac{1}{n}}{x} - \frac{-\log x}{\left(x \cdot n\right) \cdot n}\right))_*\\ \end{array}\]

Error

Derivation

`if n < -32555399.34096443 or 159200.81643882472 < n`

`if -32555399.34096443 < n < -1.61007232677794e-310`

`if -1.61007232677794e-310 < n < 159200.81643882472`

Reproduce