Average Error: 32.9 → 23.2
Time: 1.1m
Precision: 64
Internal Precision: 1344
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -156042720497.5645:\\ \;\;\;\;\frac{\log x}{\left(n \cdot n\right) \cdot x} + (\left(\frac{\frac{\frac{1}{x}}{x}}{n}\right) \cdot \frac{-1}{2} + \left(\frac{\frac{1}{x}}{n}\right))_*\\ \mathbf{elif}\;n \le 5.603654307874447 \cdot 10^{+21}:\\ \;\;\;\;\sqrt[3]{\sqrt[3]{\left(\left(e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right) \cdot \left(e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right)\right) \cdot \left(e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right)}} \cdot \left(\sqrt[3]{e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{\sqrt[3]{\left(\left(e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right) \cdot \left(e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right)\right) \cdot \left(e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\log x}{\left(n \cdot n\right) \cdot x} + (\left(\frac{\frac{\frac{1}{x}}{x}}{n}\right) \cdot \frac{-1}{2} + \left(\frac{\frac{1}{x}}{n}\right))_*\\ \end{array}\]

Error

Bits error versus x

Bits error versus n

Derivation

  1. Split input into 2 regimes

  2. if n < -156042720497.5645 or 5.603654307874447e+21 < n

    1. Initial program 45.1

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

    2. Initial simplification45.1

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

    3. Taylor expanded around -inf 63.0

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

    4. Simplified31.4

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

    if -156042720497.5645 < n < 5.603654307874447e+21

    1. Initial program 6.4

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

    2. Initial simplification6.4

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

    4. Applied add-exp-log6.5

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

    5. Applied pow-exp6.5

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

    6. Simplified5.6

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

    8. Applied add-cube-cbrt5.6

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

    10. Applied add-cbrt-cube5.6

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

    12. Applied add-cbrt-cube5.6

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

  4. Final simplification23.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -156042720497.5645:\\ \;\;\;\;\frac{\log x}{\left(n \cdot n\right) \cdot x} + (\left(\frac{\frac{\frac{1}{x}}{x}}{n}\right) \cdot \frac{-1}{2} + \left(\frac{\frac{1}{x}}{n}\right))_*\\ \mathbf{elif}\;n \le 5.603654307874447 \cdot 10^{+21}:\\ \;\;\;\;\sqrt[3]{\sqrt[3]{\left(\left(e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right) \cdot \left(e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right)\right) \cdot \left(e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right)}} \cdot \left(\sqrt[3]{e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{\sqrt[3]{\left(\left(e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right) \cdot \left(e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right)\right) \cdot \left(e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\log x}{\left(n \cdot n\right) \cdot x} + (\left(\frac{\frac{\frac{1}{x}}{x}}{n}\right) \cdot \frac{-1}{2} + \left(\frac{\frac{1}{x}}{n}\right))_*\\ \end{array}\]

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed 2018234 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))