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Derivation

1. Split input into 3 regimes

2. if (log (exp (- (+ (/ 1 (* x n)) (+ (/ (log x) n) 1)) (pow x (/ 1 n))))) < -7.037659089087615

   1. Initial program 18.6

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

   3. Applied add-sqr-sqrt18.6

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

   if -7.037659089087615 < (log (exp (- (+ (/ 1 (* x n)) (+ (/ (log x) n) 1)) (pow x (/ 1 n))))) < 1.4408519202694974e-233

   1. Initial program 39.9

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

   2. Taylor expanded around -inf 63.0

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

   3. Simplified21.7

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

   if 1.4408519202694974e-233 < (log (exp (- (+ (/ 1 (* x n)) (+ (/ (log x) n) 1)) (pow x (/ 1 n)))))

   1. Initial program 31.3

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

   3. Applied add-log-exp31.4

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

   4. Applied add-log-exp31.4

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

   5. Applied diff-log31.4

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

   6. Simplified31.4

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

   8. Applied add-cbrt-cube31.4

      \[\leadsto \color{blue}{\sqrt[3]{\left(\log \left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \log \left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\right) \cdot \log \left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}}\]
   9. Using strategy rm

   10. Applied sub-neg31.4

       \[\leadsto \sqrt[3]{\left(\log \left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \log \left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\right) \cdot \log \left(e^{\color{blue}{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} + \left(-{x}^{\left(\frac{1}{n}\right)}\right)}}\right)}\]

   11. Applied exp-sum31.3

       \[\leadsto \sqrt[3]{\left(\log \left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \log \left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\right) \cdot \log \color{blue}{\left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}} \cdot e^{-{x}^{\left(\frac{1}{n}\right)}}\right)}}\]

   12. Applied log-prod31.4

       \[\leadsto \sqrt[3]{\left(\log \left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \log \left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\right) \cdot \color{blue}{\left(\log \left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}}\right) + \log \left(e^{-{x}^{\left(\frac{1}{n}\right)}}\right)\right)}}\]

   13. Simplified31.4

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

4. Final simplification22.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(e^{\left(\left(\frac{\log x}{n} + 1\right) + \frac{1}{n \cdot x}\right) - {x}^{\left(\frac{1}{n}\right)}}\right) \le -7.037659089087615:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \sqrt{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt{{x}^{\left(\frac{1}{n}\right)}}\\ \mathbf{elif}\;\log \left(e^{\left(\left(\frac{\log x}{n} + 1\right) + \frac{1}{n \cdot x}\right) - {x}^{\left(\frac{1}{n}\right)}}\right) \le 1.4408519202694974 \cdot 10^{-233}:\\ \;\;\;\;\frac{\log x}{n \cdot \left(n \cdot x\right)} + \left(\frac{\frac{1}{n}}{x} - \frac{\frac{\frac{1}{2}}{n}}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(\log \left(e^{-{x}^{\left(\frac{1}{n}\right)}}\right) + {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right) \cdot \left(\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\right)}\\ \end{array}\]

Runtime

Time bar (total: 4.0m)Debug logProfile

herbie shell --seed 2018215 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))