Average Error: 32.2 → 1.8
Time: 30.9s
Precision: 64
Internal precision: 1408
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
⬇
\[\begin{array}{l} \mathbf{if}\;n \le -5.1490478911182765 \cdot 10^{-24}:\\ \;\;\;\;\frac{1}{x \cdot n} - \left(\frac{\frac{\frac{1}{2}}{x}}{x \cdot n} + \frac{\frac{\log x}{x}}{{n}^2}\right)\\ \mathbf{if}\;n \le 33486561961107.625:\\ \;\;\;\;{e}^{\left(\sqrt[3]{{\left(\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right)}^3}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot n} - \left(\frac{\frac{\frac{1}{2}}{x}}{x \cdot n} + \frac{\frac{\log x}{x}}{{n}^2}\right)\\ \end{array}\]

Error

Bits error versus x

Bits error versus n

Derivation

  1. Split input into 2 regimes.

  2. if n < -5.1490478911182765e-24 or 33486561961107.625 < n

    1. Initial program 44.2

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

    2. Applied taylor 8.9

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

    3. Taylor expanded around inf 8.9

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

    4. Applied simplify 1.0

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

    if -5.1490478911182765e-24 < n < 33486561961107.625

    1. Initial program 3.6

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

    3. Applied add-exp-log 3.6

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

    5. Applied add-cbrt-cube 3.6

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

    7. Applied pow1 3.6

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

    8. Applied log-pow 3.6

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

    9. Applied cube-prod 3.6

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

    10. Applied cbrt-prod 3.6

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

    11. Applied exp-prod 3.6

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

    12. Applied simplify 3.6

      \[\leadsto {\color{blue}{e}}^{\left(\sqrt[3]{{\left(\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right)}^3}\right)}\]
  3. Recombined 2 regimes into one program.
  4. Removed slow pow expressions

Runtime

Time bar (total: 30.9s) Debug logProfile

Please include this information when filing a bug report:

herbie shell --seed '#(1066500295 745726447 3908002351 725592315 4114972361 2368915013)'
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))