Average Error: 31.0 → 3.6
Time: 29.3s
Precision: 64
Internal precision: 1408
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
⬇
\[\begin{array}{l} \mathbf{if}\;n \le -4.930770214232589 \cdot 10^{+18}:\\ \;\;\;\;\frac{1}{x \cdot n} - \left(\frac{\frac{\frac{1}{2}}{x}}{x \cdot n} + \frac{\frac{\log x}{x}}{{n}^2}\right)\\ \mathbf{if}\;n \le 8.04578740725892 \cdot 10^{+26}:\\ \;\;\;\;{\left(\sqrt[3]{{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}^3}\right)}^3\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot n} - \left(\frac{\frac{\frac{1}{2}}{x}}{x \cdot n} + \frac{\frac{\log x}{x}}{{n}^2}\right)\\ \end{array}\]

Error

Bits error versus x

Bits error versus n

Derivation

  1. Split input into 2 regimes.

  2. if n < -4.930770214232589e+18 or 8.04578740725892e+26 < n

    1. Initial program 43.2

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

    2. Applied taylor 9.2

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

    3. Taylor expanded around inf 9.2

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

    4. Applied simplify 0.9

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

    if -4.930770214232589e+18 < n < 8.04578740725892e+26

    1. Initial program 8.5

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

    3. Applied add-cube-cbrt 8.5

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

    5. Applied add-cube-cbrt 8.5

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

Runtime

Time bar (total: 29.3s) Debug log

Please include this information when filing a bug report:

herbie shell --seed '#(3052192724 3812927732 3686175817 630908657 2373248591 511094450)'
(FPCore (x n)
  :name "NMSE problem 3.4.6"
  :pre (>= x 0)
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))