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Derivation

1. Split input into 3 regimes

2. if (/ 1 n) < -0.3776171992030802

   1. Initial program 0.3

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

   2. Initial simplification0.3

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

   4. Applied add-sqr-sqrt0.3

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

   5. Taylor expanded around 0 0.3

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

   7. Applied *-un-lft-identity0.3

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

   8. Applied exp-prod0.2

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

   9. Simplified0.2

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

   if -0.3776171992030802 < (/ 1 n) < 9.291941940433358e-07

   1. Initial program 44.6

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

   2. Initial simplification44.6

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

   4. Applied add-sqr-sqrt44.7

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

   5. Taylor expanded around 0 44.7

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

   6. Taylor expanded around -inf 63.0

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

   7. Simplified32.2

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

   if 9.291941940433358e-07 < (/ 1 n)

   1. Initial program 5.7

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

   2. Initial simplification5.7

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

   4. Applied pow-to-exp5.7

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

   6. Applied add-log-exp5.8

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

4. Final simplification23.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -0.3776171992030802:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \sqrt{{e}^{\left(\frac{\log x}{n}\right)}} \cdot \sqrt{{x}^{\left(\frac{1}{n}\right)}}\\ \mathbf{elif}\;\frac{1}{n} \le 9.291941940433358 \cdot 10^{-07}:\\ \;\;\;\;\frac{\frac{\frac{-1}{2}}{x}}{x \cdot n} + \left(\frac{\log x}{\left(x \cdot n\right) \cdot n} + \frac{\frac{1}{x}}{n}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - e^{\frac{1}{n} \cdot \log x}}\right)\\ \end{array}\]

Runtime

Time bar (total: 37.5s)Debug logProfile

herbie shell --seed 2018290 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))