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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -1.5177582459168592 \cdot 10^{-21}:\\ \;\;\;\;\sqrt[3]{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right) \cdot \left(\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right)}\\ \mathbf{elif}\;\frac{1}{n} \le 4.6661266554976205 \cdot 10^{-14}:\\ \;\;\;\;(\left(\frac{1}{x \cdot n}\right) \cdot \left(\frac{-\frac{1}{2}}{x}\right) + \left(\frac{1}{x \cdot n}\right))_* + \frac{\frac{\log x}{x \cdot n}}{n}\\ \mathbf{else}:\\ \;\;\;\;(e^{\log_* (1 + \left(e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right))} - 1)^*\\ \end{array}\]

Derivation

Split input into 3 regimes
if (/ 1 n) < -1.5177582459168592e-21
1. Initial program 4.8
  \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
2. Using strategy rm
3. Applied add-cbrt-cube4.8
  \[\leadsto \color{blue}{\sqrt[3]{\left(\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}}\]
if -1.5177582459168592e-21 < (/ 1 n) < 4.6661266554976205e-14
1. Initial program 44.5
  \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
2. Taylor expanded around inf 32.2
  \[\leadsto \color{blue}{\frac{1}{x \cdot n} - \left(\frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}} + \frac{1}{2} \cdot \frac{1}{{x}^{2} \cdot n}\right)}\]
3. Simplified32.1
  \[\leadsto \color{blue}{(\left(\frac{1}{n \cdot x}\right) \cdot \left(-\frac{\frac{1}{2}}{x}\right) + \left(\frac{1}{n \cdot x}\right))_* + \frac{\frac{\log x}{n \cdot x}}{n}}\]
if 4.6661266554976205e-14 < (/ 1 n)
1. Initial program 8.1
  \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
2. Using strategy rm
3. Applied add-exp-log8.2
  \[\leadsto {\color{blue}{\left(e^{\log \left(x + 1\right)}\right)}}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
4. Applied pow-exp8.2
  \[\leadsto \color{blue}{e^{\log \left(x + 1\right) \cdot \frac{1}{n}}} - {x}^{\left(\frac{1}{n}\right)}\]
5. Simplified5.1
  \[\leadsto e^{\color{blue}{\frac{\log_* (1 + x)}{n}}} - {x}^{\left(\frac{1}{n}\right)}\]
6. Using strategy rm
7. Applied expm1-log1p-u5.1
  \[\leadsto \color{blue}{(e^{\log_* (1 + \left(e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right))} - 1)^*}\]
Recombined 3 regimes into one program.
Final simplification23.5
\[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -1.5177582459168592 \cdot 10^{-21}:\\ \;\;\;\;\sqrt[3]{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right) \cdot \left(\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right)}\\ \mathbf{elif}\;\frac{1}{n} \le 4.6661266554976205 \cdot 10^{-14}:\\ \;\;\;\;(\left(\frac{1}{x \cdot n}\right) \cdot \left(\frac{-\frac{1}{2}}{x}\right) + \left(\frac{1}{x \cdot n}\right))_* + \frac{\frac{\log x}{x \cdot n}}{n}\\ \mathbf{else}:\\ \;\;\;\;(e^{\log_* (1 + \left(e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right))} - 1)^*\\ \end{array}\]

Error

Derivation

`if (/ 1 n) < -1.5177582459168592e-21`

`if -1.5177582459168592e-21 < (/ 1 n) < 4.6661266554976205e-14`

`if 4.6661266554976205e-14 < (/ 1 n)`

Runtime