Average Error: 30.9 → 13.3
Time: 1.9m
Precision: 64
Ground Truth: 128
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
⬇
\[\begin{array}{l} \mathbf{if}\;n \le -31.329673016395393:\\ \;\;\;\;\frac{1}{n \cdot x} - \left(\frac{\log x}{{n}^2 \cdot x} + \frac{1}{2} \cdot \frac{1}{\log \left(e^{n \cdot {x}^2}\right)}\right)\\ \mathbf{if}\;n \le 5667669302503732.0:\\ \;\;\;\;{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}^3\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x} - \left(\frac{\log x}{{n}^2 \cdot x} + \frac{1}{2} \cdot \frac{1}{\log \left(e^{n \cdot {x}^2}\right)}\right)\\ \end{array}\]

Error

Bits error versus x

Bits error versus n

Derivation

  1. Split input into 2 regimes.

  2. if n < -31.329673016395393 or 5667669302503732.0 < n

    1. Initial program 43.9

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

    2. Applied taylor 17.9

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

    3. Taylor expanded around inf 17.9

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

    5. Applied add-log-exp 18.0

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

    if -31.329673016395393 < n < 5667669302503732.0

    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

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

Runtime

Total time: 1.9m Debug log

Please include this information when filing a bug report:

herbie --seed '#(1317229083 385665165 927243142 4133899538 3082641653 3120456537)'
(FPCore (x n)
  :name "NMSE problem 3.4.6"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))