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Derivation

1. Split input into 3 regimes

2. if n < -1729540547.0181596

   1. Initial program 44.7

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

   2. Initial simplification44.7

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

   3. Taylor expanded around inf 32.9

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

   4. Simplified32.8

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

   6. Applied associate-/r*32.2

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

   8. Applied div-inv32.2

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

   if -1729540547.0181596 < n < 182.56083891553433

   1. Initial program 8.1

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

   2. Initial simplification8.1

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

   4. Applied *-un-lft-identity8.1

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

   5. Applied *-un-lft-identity8.1

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

   6. Applied distribute-lft-out--8.1

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

   if 182.56083891553433 < n

   1. Initial program 44.2

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

   2. Initial simplification44.2

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

   3. Taylor expanded around inf 32.6

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

   4. Simplified32.5

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

   6. Applied associate-/r*31.9

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

   8. Applied add-log-exp32.0

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

4. Final simplification22.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -1729540547.0181596:\\ \;\;\;\;\left(\frac{\frac{\frac{-1}{2}}{x}}{x \cdot n} + \frac{1}{x} \cdot \frac{1}{n}\right) + \frac{\log x}{\left(x \cdot n\right) \cdot n}\\ \mathbf{elif}\;n \le 182.56083891553433:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\log \left(e^{\frac{\frac{\frac{-1}{2}}{x}}{x \cdot n}}\right) + \frac{\frac{1}{n}}{x}\right) + \frac{\log x}{\left(x \cdot n\right) \cdot n}\\ \end{array}\]

Runtime

Time bar (total: 26.3s)Debug logProfile

herbie shell --seed 2018354 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))