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Derivation

1. Split input into 4 regimes

2. if n < -1.847675679304028e+22

   1. Initial program 45.1

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

   2. Taylor expanded around inf 32.1

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

   3. Simplified32.0

      \[\leadsto \color{blue}{\left(\frac{1}{n \cdot x} - \frac{\frac{\frac{1}{2}}{x}}{n \cdot x}\right) + \frac{\frac{\log x}{n \cdot n}}{x}}\]

   if -1.847675679304028e+22 < n < -6.33257455003563e-310

   1. Initial program 2.9

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
   2. Using strategy rm

   3. Applied add-log-exp3.1

      \[\leadsto \color{blue}{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}\]

   if -6.33257455003563e-310 < n < 10838.751390580817

   1. Initial program 25.3

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
   2. Using strategy rm

   3. Applied add-log-exp25.3

      \[\leadsto \color{blue}{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}\]
   4. Using strategy rm

   5. Applied add-cube-cbrt25.3

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

   6. Applied exp-prod25.3

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

   7. Applied log-pow25.3

      \[\leadsto \color{blue}{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \log \left(e^{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)}\]
   8. Using strategy rm

   9. Applied add-exp-log25.3

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

   10. Applied pow-exp25.3

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

   11. Simplified3.1

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

   if 10838.751390580817 < n

   1. Initial program 45.1

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
   2. Using strategy rm

   3. Applied add-log-exp45.2

      \[\leadsto \color{blue}{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}\]

   4. Taylor expanded around -inf 62.7

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

   5. Simplified32.7

      \[\leadsto \color{blue}{\frac{\frac{\frac{-1}{2}}{x}}{x \cdot n} + \left(\left(\frac{\frac{1}{x}}{n} + 0\right) + \frac{\log x}{n \cdot \left(x \cdot n\right)}\right)}\]
3. Recombined 4 regimes into one program.

4. Final simplification19.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -1.847675679304028 \cdot 10^{+22}:\\ \;\;\;\;\left(\frac{1}{x \cdot n} - \frac{\frac{\frac{1}{2}}{x}}{x \cdot n}\right) + \frac{\frac{\log x}{n \cdot n}}{x}\\ \mathbf{elif}\;n \le -6.33257455003563 \cdot 10^{-310}:\\ \;\;\;\;\log \left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\\ \mathbf{elif}\;n \le 10838.751390580817:\\ \;\;\;\;\sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \log \left(e^{\sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{-1}{2}}{x}}{x \cdot n} + \left(\frac{\log x}{n \cdot \left(x \cdot n\right)} + \frac{\frac{1}{x}}{n}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019002 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))

Details

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