Error

Derivation

1. Split input into 3 regimes

2. if (/ 1 n) < -1.1924281163064566e-07

   1. Initial program 0.9

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

   3. Applied add-cube-cbrt0.9

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

   4. Applied unpow-prod-down1.0

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

   5. Applied fma-neg1.0

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

   6. Taylor expanded around -inf 63.0

      \[\leadsto (\left({\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)}\right) \cdot \left({\color{blue}{\left(\left(e^{\frac{1}{3} \cdot \left(\log -1 - \log \left(\frac{-1}{x}\right)\right)} + \frac{1}{3} \cdot \frac{e^{\frac{1}{3} \cdot \left(\log -1 - \log \left(\frac{-1}{x}\right)\right)}}{x}\right) - \frac{1}{9} \cdot \frac{e^{\frac{1}{3} \cdot \left(\log -1 - \log \left(\frac{-1}{x}\right)\right)}}{{x}^{2}}\right)}}^{\left(\frac{1}{n}\right)}\right) + \left(-{x}^{\left(\frac{1}{n}\right)}\right))_*\]

   7. Simplified1.6

      \[\leadsto (\left({\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)}\right) \cdot \left({\color{blue}{\left((\left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right) \cdot \left(\frac{\sqrt[3]{x}}{x}\right) + \left(\sqrt[3]{x}\right))_*\right)}}^{\left(\frac{1}{n}\right)}\right) + \left(-{x}^{\left(\frac{1}{n}\right)}\right))_*\]

   if -1.1924281163064566e-07 < (/ 1 n) < 9.66942257384201e-07

   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 63.0

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

   3. Simplified31.4

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

   if 9.66942257384201e-07 < (/ 1 n)

   1. Initial program 25.6

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

   3. Applied add-exp-log25.6

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

   4. Applied pow-exp25.6

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

   5. Simplified1.6

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

4. Final simplification18.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -1.1924281163064566 \cdot 10^{-07}:\\ \;\;\;\;(\left({\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)}\right) \cdot \left({\left((\left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right) \cdot \left(\frac{\sqrt[3]{x}}{x}\right) + \left(\sqrt[3]{x}\right))_*\right)}^{\left(\frac{1}{n}\right)}\right) + \left(-{x}^{\left(\frac{1}{n}\right)}\right))_*\\ \mathbf{elif}\;\frac{1}{n} \le 9.66942257384201 \cdot 10^{-07}:\\ \;\;\;\;\left(\frac{\frac{1}{x}}{n} + \frac{\log x}{n \cdot \left(x \cdot n\right)}\right) + \frac{\frac{\frac{-1}{2}}{x}}{x \cdot n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019004 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))

Details

Time bar (total: 31.6s)Debug log