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Derivation

1. Split input into 4 regimes

2. if n < -1803761.66460632

   1. Initial program 44.5

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

   3. Applied add-exp-log44.5

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

   4. Applied pow-exp44.5

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

   5. Simplified44.5

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

   6. Taylor expanded around -inf 63.2

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

   7. Simplified32.0

      \[\leadsto \color{blue}{\frac{\frac{\frac{-1}{2}}{x}}{x \cdot n} + \left(\left(\frac{\frac{1}{x}}{n} + 0\right) + \frac{\log x}{n \cdot \left(x \cdot n\right)}\right)}\]
   8. Using strategy rm

   9. Applied add-sqr-sqrt32.1

      \[\leadsto \frac{\frac{\frac{-1}{2}}{x}}{x \cdot n} + \left(\left(\frac{\color{blue}{\sqrt{\frac{1}{x}} \cdot \sqrt{\frac{1}{x}}}}{n} + 0\right) + \frac{\log x}{n \cdot \left(x \cdot n\right)}\right)\]

   10. Applied associate-/l*32.1

       \[\leadsto \frac{\frac{\frac{-1}{2}}{x}}{x \cdot n} + \left(\left(\color{blue}{\frac{\sqrt{\frac{1}{x}}}{\frac{n}{\sqrt{\frac{1}{x}}}}} + 0\right) + \frac{\log x}{n \cdot \left(x \cdot n\right)}\right)\]

   if -1803761.66460632 < n < -5.6048574067154e-308

   1. Initial program 0.5

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

   if -5.6048574067154e-308 < n < 1716508256.237214

   1. Initial program 25.8

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

   3. Applied add-exp-log25.9

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

   4. Applied pow-exp25.9

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

   5. Simplified1.8

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

   if 1716508256.237214 < n

   1. Initial program 44.2

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

   3. Applied add-exp-log44.2

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

   4. Applied pow-exp44.2

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

   5. Simplified44.2

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

   6. Taylor expanded around -inf 62.7

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

   7. Simplified31.4

      \[\leadsto \color{blue}{\frac{\frac{\frac{-1}{2}}{x}}{x \cdot n} + \left(\left(\frac{\frac{1}{x}}{n} + 0\right) + \frac{\log x}{n \cdot \left(x \cdot n\right)}\right)}\]
   8. Using strategy rm

   9. Applied add-log-exp31.4

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

4. Final simplification18.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -1803761.66460632:\\ \;\;\;\;\left(\frac{\sqrt{\frac{1}{x}}}{\frac{n}{\sqrt{\frac{1}{x}}}} + \frac{\log x}{n \cdot \left(x \cdot n\right)}\right) + \frac{\frac{\frac{-1}{2}}{x}}{x \cdot n}\\ \mathbf{elif}\;n \le -5.6048574067154 \cdot 10^{-308}:\\ \;\;\;\;{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;n \le 1716508256.237214:\\ \;\;\;\;e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\log x}{n \cdot \left(x \cdot n\right)} + \frac{\frac{1}{x}}{n}\right) + \log \left(e^{\frac{\frac{\frac{-1}{2}}{x}}{x \cdot n}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019008 +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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