Error

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. Taylor expanded around -inf 63.2

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

   3. Simplified32.2

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

   5. Applied flip-+57.7

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

   6. Applied frac-add59.4

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

   7. Simplified55.2

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

   8. Simplified32.2

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

   10. Applied add-cube-cbrt32.2

       \[\leadsto \frac{\frac{\frac{-1}{2}}{x}}{x \cdot n} + \frac{(\left(\log x\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{\frac{1}{x}}{n}} \cdot \sqrt[3]{\frac{\frac{1}{x}}{n}}\right) \cdot \sqrt[3]{\frac{\frac{1}{x}}{n}}\right)} + \left(n \cdot \frac{\frac{1}{x}}{n}\right))_*}{n}\]

   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. Using strategy rm

   3. Applied add-sqr-sqrt8.1

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

   4. Applied fma-neg8.1

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

   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. Taylor expanded around -inf 62.7

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

   3. Simplified31.9

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

   5. Applied flip-+57.4

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

   6. Applied frac-add58.9

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

   7. Simplified54.2

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

   8. Simplified31.9

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

   10. Applied add-log-exp32.0

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

4. Final simplification22.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -1729540547.0181596:\\ \;\;\;\;\frac{(\left(\log x\right) \cdot \left(\left(\sqrt[3]{\frac{\frac{1}{x}}{n}} \cdot \sqrt[3]{\frac{\frac{1}{x}}{n}}\right) \cdot \sqrt[3]{\frac{\frac{1}{x}}{n}}\right) + \left(\frac{\frac{1}{x}}{n} \cdot n\right))_*}{n} + \frac{\frac{\frac{-1}{2}}{x}}{x \cdot n}\\ \mathbf{elif}\;n \le 182.56083891553433:\\ \;\;\;\;(\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right) + \left(-{x}^{\left(\frac{1}{n}\right)}\right))_*\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left(\log x\right) \cdot \left(\frac{\frac{1}{x}}{n}\right) + \left(\frac{\frac{1}{x}}{n} \cdot n\right))_*}{n} + \log \left(e^{\frac{\frac{\frac{-1}{2}}{x}}{x \cdot n}}\right)\\ \end{array}\]

Runtime

Time bar (total: 31.5s)Debug logProfile

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