Error

Derivation

1. Split input into 3 regimes

2. if n < -66858541040.3067

   1. Initial program 45.4

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

   2. Initial simplification45.4

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

   4. Applied add-exp-log45.4

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

   5. Applied pow-exp45.4

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

   6. Simplified45.4

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

   8. Applied add-cube-cbrt45.4

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

   9. Applied fma-neg45.4

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

   10. Taylor expanded around -inf 63.2

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

   11. Simplified31.6

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

   13. Applied add-log-exp31.7

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

   if -66858541040.3067 < n < 140898664.52800614

   1. Initial program 3.7

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

   2. Initial simplification3.7

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

   4. Applied add-exp-log3.8

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

   5. Applied pow-exp3.7

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

   6. Simplified2.7

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

   8. Applied add-cube-cbrt2.7

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

   9. Applied fma-neg2.7

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

   if 140898664.52800614 < n

   1. Initial program 45.3

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

   2. Initial simplification45.3

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

   4. Applied add-exp-log45.3

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

   5. Applied pow-exp45.3

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

   6. Simplified45.3

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

   8. Applied add-cube-cbrt45.3

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

   9. Applied fma-neg45.3

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

   10. Taylor expanded around inf 33.0

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

   11. Simplified32.9

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

4. Final simplification23.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -66858541040.3067:\\ \;\;\;\;\log \left(e^{\frac{\frac{\frac{-1}{2}}{x}}{x \cdot n}}\right) + \left(\frac{\frac{1}{x}}{n} + \frac{\log x}{n \cdot \left(x \cdot n\right)}\right)\\ \mathbf{elif}\;n \le 140898664.52800614:\\ \;\;\;\;(\left(\sqrt[3]{e^{\frac{\log_* (1 + x)}{n}}} \cdot \sqrt[3]{e^{\frac{\log_* (1 + x)}{n}}}\right) \cdot \left(\sqrt[3]{e^{\frac{\log_* (1 + x)}{n}}}\right) + \left(-{x}^{\left(\frac{1}{n}\right)}\right))_*\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\log x}{n \cdot n}}{x} + \left(\frac{1}{x \cdot n} - \frac{\frac{\frac{1}{2}}{x}}{x \cdot n}\right)\\ \end{array}\]

Runtime

Time bar (total: 46.1s)Debug logProfile

herbie shell --seed 2018252 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))