Error

Try it out

Derivation

1. Split input into 2 regimes

2. if n < -1150938376.394078 or 1049.9430934270722 < n

   1. Initial program 44.5

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

   2. Initial simplification44.5

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

   3. Taylor expanded around -inf 63.0

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

   4. Simplified31.1

      \[\leadsto \color{blue}{\frac{\frac{\frac{-1}{2}}{x}}{x \cdot n} + \left(\left(\frac{\frac{1}{x}}{n} + 0\right) + \frac{\log x}{n \cdot \left(x \cdot n\right)}\right)}\]

   if -1150938376.394078 < n < 1049.9430934270722

   1. Initial program 2.4

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

   2. Initial simplification2.4

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

   4. Applied add-exp-log2.5

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

   5. Applied pow-exp2.5

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

   6. Simplified1.4

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

   8. Applied add-log-exp1.7

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

   9. Applied add-log-exp1.9

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

   10. Applied diff-log1.9

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

   11. Simplified1.9

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

   13. Applied add-sqr-sqrt1.9

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

   14. Applied log-prod1.9

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

4. Final simplification22.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -1150938376.394078 \lor \neg \left(n \le 1049.9430934270722\right):\\ \;\;\;\;\frac{\frac{\frac{-1}{2}}{x}}{x \cdot n} + \left(\frac{\log x}{n \cdot \left(x \cdot n\right)} + \frac{\frac{1}{x}}{n}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{e^{e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}}}\right) + \log \left(\sqrt{e^{e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}}}\right)\\ \end{array}\]

Runtime

Time bar (total: 41.3s)Debug logProfile

herbie shell --seed 2018339 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))