Average Error: 31.3 → 3.0
Time: 28.1s
Precision: 64
Internal precision: 1408
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
⬇
\[\begin{array}{l} \mathbf{if}\;n \le -934654479866939.4:\\ \;\;\;\;\frac{1}{x \cdot n} - \left(\frac{\frac{\frac{1}{2}}{x}}{x \cdot n} + \frac{\frac{\log x}{x}}{{n}^2}\right)\\ \mathbf{if}\;n \le 1.4682660339097687 \cdot 10^{+21}:\\ \;\;\;\;{\left(\sqrt[3]{{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}^3}\right)}^3\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot n} - \left(\frac{\frac{\frac{1}{2}}{x}}{x \cdot n} + \frac{\frac{\log x}{x}}{{n}^2}\right)\\ \end{array}\]

Error

Bits error versus x

Bits error versus n

Derivation

  1. Split input into 2 regimes.

  2. if n < -934654479866939.4 or 1.4682660339097687e+21 < n

    1. Initial program 43.9

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

    2. Applied taylor 9.3

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

    3. Taylor expanded around inf 9.3

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

    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 -934654479866939.4 < n < 1.4682660339097687e+21

    1. Initial program 6.8

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

    3. Applied add-cube-cbrt 6.8

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

    5. Applied add-cube-cbrt 6.8

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

Runtime

Time bar (total: 28.1s) Debug logProfile

Please include this information when filing a bug report:

herbie shell --seed '#(1067773715 2765207660 218871639 3688798924 2755544087 2054563380)'
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))