Average Error: 30.8 → 6.7
Time: 2.9m
Precision: 64
Ground Truth: 128
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
⬇
\[\begin{array}{l} \mathbf{if}\;n \le -961281.8751595157:\\ \;\;\;\;\left(\frac{\frac{1}{n}}{x} - \frac{\frac{\frac{1}{2}}{n}}{{x}^2}\right) - \frac{\log x}{x \cdot {n}^2}\\ \mathbf{if}\;n \le 10199713985.948484:\\ \;\;\;\;{\left({\left(\sqrt[3]{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)}^3\right)}^3\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{1}{n}}{x} - \frac{\frac{\frac{1}{2}}{n}}{{x}^2}\right) - \frac{\log x}{x \cdot {n}^2}\\ \end{array}\]

Error

Bits error versus x

Bits error versus n

Derivation

  1. Split input into 2 regimes.

  2. if n < -961281.8751595157 or 10199713985.948484 < n

    1. Initial program 43.9

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

    3. Applied add-cube-cbrt 43.9

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

    5. Applied add-log-exp 43.9

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

    6. Applied add-log-exp 43.9

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

    7. Applied diff-log 43.9

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

    8. Applied simplify 43.9

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

    9. Applied taylor 42.6

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

    10. Taylor expanded around inf 42.6

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

    11. Applied simplify 8.3

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

    12. Applied simplify 8.3

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

    if -961281.8751595157 < n < 10199713985.948484

    1. Initial program 3.5

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

    3. Applied add-cube-cbrt 3.5

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

    5. Applied add-cube-cbrt 3.5

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

Runtime

Total time: 2.9m Debug log

Please include this information when filing a bug report:

herbie --seed '#(4266650265 2609020814 1375209603 2731663559 2960833714 1594438513)'
(FPCore (x n)
  :name "NMSE problem 3.4.6"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))