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Derivation

1. Split input into 3 regimes

2. if n < -206.4372690337579

   1. Initial program 44.3

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

   2. Initial simplification44.3

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

   4. Applied add-exp-log44.3

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

   5. Taylor expanded around inf 32.9

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

   6. Simplified32.8

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

   8. Applied associate-/r*32.2

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

   10. Applied add-log-exp32.2

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

   if -206.4372690337579 < n < 5.23549665628974e+44

   1. Initial program 11.9

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

   2. Initial simplification11.9

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

   4. Applied add-exp-log11.9

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

   if 5.23549665628974e+44 < n

   1. Initial program 43.4

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

   2. Initial simplification43.4

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

   4. Applied add-exp-log43.4

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

   5. Taylor expanded around inf 32.0

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

   6. Simplified32.0

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

   8. Applied associate-/r*31.4

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

   10. Applied div-inv31.4

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

   11. Applied *-un-lft-identity31.4

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

   12. Applied times-frac31.5

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

   13. Applied distribute-lft-out31.5

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

4. Final simplification22.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -206.4372690337579:\\ \;\;\;\;\frac{\log x}{\left(x \cdot n\right) \cdot n} + \left(\frac{\frac{1}{n}}{x} + \log \left(e^{\frac{\frac{\frac{-1}{2}}{x}}{x \cdot n}}\right)\right)\\ \mathbf{elif}\;n \le 5.23549665628974 \cdot 10^{+44}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - e^{\log \left({x}^{\left(\frac{1}{n}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x} + \frac{\frac{\frac{-1}{2}}{x}}{x}\right) \cdot \frac{1}{n} + \frac{\log x}{\left(x \cdot n\right) \cdot n}\\ \end{array}\]

Runtime

Time bar (total: 30.5s)Debug logProfile

herbie shell --seed 2018351 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))