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Derivation

1. Split input into 2 regimes

2. if n < -5696300804586787.0 or 14646138553292.576 < 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. Using strategy rm

   7. Applied add-sqr-sqrt44.2

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

   8. Applied add-sqr-sqrt44.2

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

   9. Applied difference-of-squares44.2

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

   10. Taylor expanded around -inf 63.0

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

   11. Simplified31.3

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

   if -5696300804586787.0 < n < 14646138553292.576

   1. Initial program 5.3

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

   3. Applied add-exp-log5.3

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

   4. Applied pow-exp5.3

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

   5. Simplified4.3

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

   7. Applied add-log-exp4.8

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

   8. Applied add-log-exp4.8

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

   9. Applied diff-log4.9

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

   10. Simplified4.8

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

4. Final simplification23.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -5696300804586787.0 \lor \neg \left(n \le 14646138553292.576\right):\\ \;\;\;\;\frac{\frac{\frac{-1}{2}}{x}}{x \cdot n} + \left(\frac{\log x}{n \cdot \left(x \cdot n\right)} + \frac{\frac{1}{x}}{n}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}}\right)\\ \end{array}\]

Runtime

Time bar (total: 1.5m)Debug logProfile

herbie shell --seed 2018286 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))