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Derivation

1. Split input into 2 regimes

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

   1. Initial program 2.2

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

   2. Initial simplification2.2

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

   4. Applied add-sqr-sqrt2.2

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

   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. Taylor expanded around 0 44.6

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

   5. Applied *-un-lft-identity44.6

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

   6. Applied exp-prod44.6

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

   7. Simplified44.6

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

   8. Taylor expanded around inf 32.8

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

   9. Simplified32.8

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

4. Final simplification24.1

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

Runtime

Time bar (total: 29.6s)Debug logProfile

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