Average Error: 31.6 → 6.8
Time: 2.4m
Precision: 64
Internal precision: 1408
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
⬇
\[\begin{array}{l} \mathbf{if}\;n \le -2.672258838309128 \cdot 10^{-11}:\\ \;\;\;\;\left(\frac{\frac{1}{x}}{n} - \frac{\frac{\frac{1}{2}}{n}}{{x}^2}\right) - \frac{\log x}{n \cdot \left(n \cdot x\right)}\\ \mathbf{if}\;n \le 19699878403.887928:\\ \;\;\;\;e^{{\left(\sqrt[3]{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}\right)}^3}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{1}{x}}{n} - \frac{\frac{\frac{1}{2}}{n}}{{x}^2}\right) - \frac{\log x}{n \cdot \left(n \cdot x\right)}\\ \end{array}\]

Error

Bits error versus x

Bits error versus n

Derivation

  1. Split input into 2 regimes.

  2. if n < -2.672258838309128e-11 or 19699878403.887928 < n

    1. Initial program 44.2

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

    3. Applied add-exp-log 44.2

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

    4. Applied taylor 29.6

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

    5. Taylor expanded around inf 29.6

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

    6. Applied simplify 0.3

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

    7. Applied simplify 8.6

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

    if -2.672258838309128e-11 < n < 19699878403.887928

    1. Initial program 2.7

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

    3. Applied add-exp-log 2.7

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

    5. Applied add-cube-cbrt 2.7

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

Runtime

Time bar (total: 2.4m) Debug log

Please include this information when filing a bug report:

herbie shell --seed '#(772101555 1905824529 294602591 2478279198 2123125427 4197813737)'
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))