Average Error: 32.8 → 23.9
Time: 41.2s
Precision: 64
Internal Precision: 1344
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -124019476395.04863 \lor \neg \left(n \le 140898664.52800614\right):\\ \;\;\;\;\left(\frac{1}{x \cdot n} + \frac{\frac{\frac{-1}{2}}{x}}{x \cdot n}\right) + \frac{\log x}{\left(x \cdot n\right) \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} \cdot \left(\sqrt[3]{e^{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}} \cdot \sqrt[3]{e^{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array}\]

Error

Bits error versus x

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if n < -124019476395.04863 or 140898664.52800614 < n

    1. Initial program 45.3

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

    2. Initial simplification45.3

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

    4. Applied add-cube-cbrt45.4

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

    6. Applied add-exp-log45.4

      \[\leadsto \left(\sqrt[3]{\color{blue}{e^{\log \left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)}\right)}}} \cdot \sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{1}{n}\right)}\]

    7. Taylor expanded around inf 32.6

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

    8. Simplified32.6

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

    if -124019476395.04863 < n < 140898664.52800614

    1. Initial program 3.7

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

    2. Initial simplification3.7

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

    4. Applied add-cube-cbrt3.8

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

    6. Applied add-exp-log3.8

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

    8. Applied add-exp-log3.8

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

  4. Final simplification23.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -124019476395.04863 \lor \neg \left(n \le 140898664.52800614\right):\\ \;\;\;\;\left(\frac{1}{x \cdot n} + \frac{\frac{\frac{-1}{2}}{x}}{x \cdot n}\right) + \frac{\log x}{\left(x \cdot n\right) \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} \cdot \left(\sqrt[3]{e^{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}} \cdot \sqrt[3]{e^{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array}\]

Runtime

Time bar (total: 41.2s)Debug logProfile

herbie shell --seed 2018252 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))