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Derivation

1. Split input into 3 regimes

2. if n < -1394.5787406908253

   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 32.7

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

   3. Simplified32.6

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

   5. Applied associate-/r*31.9

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

   7. Applied div-inv31.9

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

   8. Applied associate-/l*32.7

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

   if -1394.5787406908253 < n < 243695.00651274333

   1. Initial program 7.9

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

   3. Applied add-sqr-sqrt7.9

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

   if 243695.00651274333 < n

   1. Initial program 44.4

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

   2. Taylor expanded around inf 32.7

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

   3. Simplified32.7

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

   5. Applied associate-/r*31.9

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

   7. Applied add-log-exp31.9

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

4. Final simplification21.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -1394.5787406908253:\\ \;\;\;\;\left(\frac{\frac{\frac{-1}{2}}{x}}{x \cdot n} + \frac{1}{\frac{x}{\frac{1}{n}}}\right) + \frac{\log x}{\left(x \cdot n\right) \cdot n}\\ \mathbf{elif}\;n \le 243695.00651274333:\\ \;\;\;\;\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log x}{\left(x \cdot n\right) \cdot n} + \left(\frac{\frac{1}{n}}{x} + \log \left(e^{\frac{\frac{\frac{-1}{2}}{x}}{x \cdot n}}\right)\right)\\ \end{array}\]

Runtime

Time bar (total: 30.0s)Debug logProfile

herbie shell --seed 2018353 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))