Error

Derivation

1. Split input into 3 regimes

2. if n < -30098.933090170118

   1. Initial program 45.4

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

   3. Applied add-exp-log45.4

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

   4. Applied pow-exp45.4

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

   5. Simplified45.4

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

   7. Applied add-sqr-sqrt45.4

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

   8. Applied add-sqr-sqrt45.4

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

   9. Applied difference-of-squares45.4

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

   10. Taylor expanded around inf 32.9

       \[\leadsto \left(\sqrt{e^{\frac{\log_* (1 + x)}{n}}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{x \cdot n} - \left(\frac{1}{4} \cdot \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}} + \frac{1}{4} \cdot \frac{1}{{x}^{2} \cdot n}\right)\right)}\]

   11. Simplified32.9

       \[\leadsto \left(\sqrt{e^{\frac{\log_* (1 + x)}{n}}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \color{blue}{(\left(\frac{\frac{\frac{1}{4}}{n}}{n}\right) \cdot \left(\frac{\log x}{x}\right) + \left(\frac{\frac{1}{2}}{x \cdot n} - \frac{\frac{\frac{1}{4}}{n}}{x \cdot x}\right))_*}\]
   12. Using strategy rm

   13. Applied add-log-exp32.9

       \[\leadsto \left(\sqrt{e^{\frac{\log_* (1 + x)}{n}}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot (\left(\frac{\frac{\frac{1}{4}}{n}}{n}\right) \cdot \left(\frac{\log x}{x}\right) + \left(\frac{\frac{1}{2}}{x \cdot n} - \color{blue}{\log \left(e^{\frac{\frac{\frac{1}{4}}{n}}{x \cdot x}}\right)}\right))_*\]

   if -30098.933090170118 < n < 133366875751.63362

   1. Initial program 3.6

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

   3. Applied add-exp-log3.6

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

   4. Applied pow-exp3.6

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

   5. Simplified2.6

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

   if 133366875751.63362 < n

   1. Initial program 44.9

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

   3. Applied add-exp-log44.9

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

   4. Applied pow-exp44.9

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

   5. Simplified44.9

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

   7. Applied add-sqr-sqrt44.9

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

   8. Applied add-sqr-sqrt44.9

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

   9. Applied difference-of-squares44.9

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

   10. Taylor expanded around inf 32.1

       \[\leadsto \left(\sqrt{e^{\frac{\log_* (1 + x)}{n}}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{x \cdot n} - \left(\frac{1}{4} \cdot \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}} + \frac{1}{4} \cdot \frac{1}{{x}^{2} \cdot n}\right)\right)}\]

   11. Simplified32.1

       \[\leadsto \left(\sqrt{e^{\frac{\log_* (1 + x)}{n}}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \color{blue}{(\left(\frac{\frac{\frac{1}{4}}{n}}{n}\right) \cdot \left(\frac{\log x}{x}\right) + \left(\frac{\frac{1}{2}}{x \cdot n} - \frac{\frac{\frac{1}{4}}{n}}{x \cdot x}\right))_*}\]
   12. Using strategy rm

   13. Applied add-sqr-sqrt32.1

       \[\leadsto \left(\sqrt{e^{\frac{\log_* (1 + x)}{n}}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot (\left(\frac{\frac{\frac{1}{4}}{n}}{n}\right) \cdot \left(\frac{\log x}{x}\right) + \left(\frac{\frac{1}{2}}{x \cdot n} - \frac{\color{blue}{\sqrt{\frac{\frac{1}{4}}{n}} \cdot \sqrt{\frac{\frac{1}{4}}{n}}}}{x \cdot x}\right))_*\]

   14. Applied times-frac32.0

       \[\leadsto \left(\sqrt{e^{\frac{\log_* (1 + x)}{n}}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot (\left(\frac{\frac{\frac{1}{4}}{n}}{n}\right) \cdot \left(\frac{\log x}{x}\right) + \left(\frac{\frac{1}{2}}{x \cdot n} - \color{blue}{\frac{\sqrt{\frac{\frac{1}{4}}{n}}}{x} \cdot \frac{\sqrt{\frac{\frac{1}{4}}{n}}}{x}}\right))_*\]

   15. Applied div-inv32.0

       \[\leadsto \left(\sqrt{e^{\frac{\log_* (1 + x)}{n}}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot (\left(\frac{\frac{\frac{1}{4}}{n}}{n}\right) \cdot \left(\frac{\log x}{x}\right) + \left(\color{blue}{\frac{1}{2} \cdot \frac{1}{x \cdot n}} - \frac{\sqrt{\frac{\frac{1}{4}}{n}}}{x} \cdot \frac{\sqrt{\frac{\frac{1}{4}}{n}}}{x}\right))_*\]

   16. Applied prod-diff31.9

       \[\leadsto \left(\sqrt{e^{\frac{\log_* (1 + x)}{n}}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot (\left(\frac{\frac{\frac{1}{4}}{n}}{n}\right) \cdot \left(\frac{\log x}{x}\right) + \color{blue}{\left((\frac{1}{2} \cdot \left(\frac{1}{x \cdot n}\right) + \left(-\frac{\sqrt{\frac{\frac{1}{4}}{n}}}{x} \cdot \frac{\sqrt{\frac{\frac{1}{4}}{n}}}{x}\right))_* + (\left(-\frac{\sqrt{\frac{\frac{1}{4}}{n}}}{x}\right) \cdot \left(\frac{\sqrt{\frac{\frac{1}{4}}{n}}}{x}\right) + \left(\frac{\sqrt{\frac{\frac{1}{4}}{n}}}{x} \cdot \frac{\sqrt{\frac{\frac{1}{4}}{n}}}{x}\right))_*\right)})_*\]

   17. Simplified31.3

       \[\leadsto \left(\sqrt{e^{\frac{\log_* (1 + x)}{n}}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot (\left(\frac{\frac{\frac{1}{4}}{n}}{n}\right) \cdot \left(\frac{\log x}{x}\right) + \left(\color{blue}{\left(\frac{\frac{\frac{1}{2}}{x}}{n} - \frac{\frac{\frac{1}{4}}{n}}{x \cdot x}\right)} + (\left(-\frac{\sqrt{\frac{\frac{1}{4}}{n}}}{x}\right) \cdot \left(\frac{\sqrt{\frac{\frac{1}{4}}{n}}}{x}\right) + \left(\frac{\sqrt{\frac{\frac{1}{4}}{n}}}{x} \cdot \frac{\sqrt{\frac{\frac{1}{4}}{n}}}{x}\right))_*\right))_*\]

   18. Simplified31.4

       \[\leadsto \left(\sqrt{e^{\frac{\log_* (1 + x)}{n}}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot (\left(\frac{\frac{\frac{1}{4}}{n}}{n}\right) \cdot \left(\frac{\log x}{x}\right) + \left(\left(\frac{\frac{\frac{1}{2}}{x}}{n} - \frac{\frac{\frac{1}{4}}{n}}{x \cdot x}\right) + \color{blue}{0}\right))_*\]
3. Recombined 3 regimes into one program.

4. Final simplification23.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -30098.933090170118:\\ \;\;\;\;(\left(\frac{\frac{\frac{1}{4}}{n}}{n}\right) \cdot \left(\frac{\log x}{x}\right) + \left(\frac{\frac{1}{2}}{x \cdot n} - \log \left(e^{\frac{\frac{\frac{1}{4}}{n}}{x \cdot x}}\right)\right))_* \cdot \left(\sqrt{{x}^{\left(\frac{1}{n}\right)}} + \sqrt{e^{\frac{\log_* (1 + x)}{n}}}\right)\\ \mathbf{elif}\;n \le 133366875751.63362:\\ \;\;\;\;e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{{x}^{\left(\frac{1}{n}\right)}} + \sqrt{e^{\frac{\log_* (1 + x)}{n}}}\right) \cdot (\left(\frac{\frac{\frac{1}{4}}{n}}{n}\right) \cdot \left(\frac{\log x}{x}\right) + \left(\frac{\frac{\frac{1}{2}}{x}}{n} - \frac{\frac{\frac{1}{4}}{n}}{x \cdot x}\right))_*\\ \end{array}\]

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed 2018258 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))