Error

Derivation

1. Split input into 3 regimes

2. if (/ 1 n) < -0.00011078710776013498

   1. Initial program 0.8

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

   3. Applied add-sqr-sqrt0.9

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

   4. Applied add-sqr-sqrt0.8

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

   5. Applied difference-of-squares0.8

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

   7. Applied add-sqr-sqrt0.8

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

   8. Applied associate-*l*0.8

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

   if -0.00011078710776013498 < (/ 1 n) < 8.049603106300568e-11

   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-sqr-sqrt45.0

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

   4. Applied add-sqr-sqrt45.0

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

   5. Applied difference-of-squares45.0

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

   6. Taylor expanded around inf 32.2

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

   7. Simplified32.1

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

   9. Applied add-log-exp32.2

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

   if 8.049603106300568e-11 < (/ 1 n)

   1. Initial program 6.2

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

   3. Applied add-sqr-sqrt6.2

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

   4. Applied add-sqr-sqrt6.2

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

   5. Applied difference-of-squares6.2

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

   7. Applied *-un-lft-identity6.2

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

   8. Applied sqrt-prod6.2

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

   9. Applied add-cube-cbrt6.2

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

   10. Applied sqrt-prod6.2

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

   11. Applied prod-diff6.2

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

   12. Applied distribute-lft-in6.2

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

   13. Simplified6.2

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

4. Final simplification23.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -0.00011078710776013498:\\ \;\;\;\;\sqrt{\sqrt{{x}^{\left(\frac{1}{n}\right)}} + \sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}} \cdot \left(\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt{\sqrt{{x}^{\left(\frac{1}{n}\right)}} + \sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}}\right)\\ \mathbf{elif}\;\frac{1}{n} \le 8.049603106300568 \cdot 10^{-11}:\\ \;\;\;\;\left(\sqrt{{x}^{\left(\frac{1}{n}\right)}} + \sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right) \cdot (\left(\frac{\frac{\frac{1}{4}}{n}}{n}\right) \cdot \left(\frac{\log x}{x}\right) + \left(\frac{\frac{1}{2}}{x \cdot n} - \log \left(e^{\frac{\frac{\frac{1}{4}}{n}}{x \cdot x}}\right)\right))_*\\ \mathbf{else}:\\ \;\;\;\;(\left(\sqrt{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}}\right) \cdot \left(\sqrt{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}}\right) + \left(-\sqrt{{x}^{\left(\frac{1}{n}\right)}}\right))_* \cdot \left(\sqrt{{x}^{\left(\frac{1}{n}\right)}} + \sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right)\\ \end{array}\]

Runtime

Time bar (total: 3.4m)Debug logProfile

herbie shell --seed 2018263 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))