Average Error: 30.9 → 7.7
Time: 48.6s
Precision: 64
Internal precision: 1408
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
⬇
\[\begin{array}{l} \mathbf{if}\;n \le -13459018702146.598:\\ \;\;\;\;\left(\frac{1}{n \cdot x} - \frac{\frac{\frac{1}{2}}{n}}{{x}^2}\right) - \frac{\frac{\log x}{x}}{{n}^2}\\ \mathbf{if}\;n \le 3663814092.5485716:\\ \;\;\;\;{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right)}^3 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{n \cdot x} - \frac{\frac{\frac{1}{2}}{n}}{{x}^2}\right) - \frac{\frac{\log x}{x}}{{n}^2}\\ \end{array}\]

Error

Bits error versus x

Bits error versus n

Derivation

  1. Split input into 2 regimes.

  2. if n < -13459018702146.598 or 3663814092.5485716 < n

    1. Initial program 43.6

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

    2. Applied taylor 9.2

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

    3. Taylor expanded around inf 9.2

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

    4. Applied simplify 0.9

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

    5. Applied simplify 9.2

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

    if -13459018702146.598 < n < 3663814092.5485716

    1. Initial program 4.6

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

    3. Applied add-cube-cbrt 4.7

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

Runtime

Time bar (total: 48.6s) Debug log

Please include this information when filing a bug report:

herbie shell --seed '#(644180380 3784176976 401987740 22459203 1940947670 3323606534)'
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))