Average Error: 33.2 → 24.0
Time: 41.5s
Precision: 64
Internal Precision: 1344
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -30098.933090170118:\\ \;\;\;\;\left(\frac{1}{x \cdot n} + \frac{\frac{\frac{-1}{2}}{x}}{x \cdot n}\right) + \frac{\log x}{\left(x \cdot n\right) \cdot n}\\ \mathbf{elif}\;n \le 0.47957363781988654:\\ \;\;\;\;\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} \cdot \left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\frac{-1}{2}}{x} + \left(\frac{\log x}{n} - \left(\log n + \log x\right)\right)}\\ \end{array}\]

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 3 regimes

  2. if n < -30098.933090170118

    1. Initial program 45.4

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

    2. Initial simplification45.4

      \[\leadsto {\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
    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-log46.1

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

    7. Taylor expanded around inf 33.0

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

    8. Simplified32.9

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

    if -30098.933090170118 < n < 0.47957363781988654

    1. Initial program 1.7

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

    2. Initial simplification1.7

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

    4. Applied add-cube-cbrt1.7

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

    if 0.47957363781988654 < n

    1. Initial program 44.6

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

    2. Initial simplification44.6

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

    4. Applied add-cube-cbrt44.6

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

    6. Applied add-exp-log44.6

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

    7. Taylor expanded around inf 31.7

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

    8. Simplified31.7

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

  4. Final simplification24.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -30098.933090170118:\\ \;\;\;\;\left(\frac{1}{x \cdot n} + \frac{\frac{\frac{-1}{2}}{x}}{x \cdot n}\right) + \frac{\log x}{\left(x \cdot n\right) \cdot n}\\ \mathbf{elif}\;n \le 0.47957363781988654:\\ \;\;\;\;\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} \cdot \left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\frac{-1}{2}}{x} + \left(\frac{\log x}{n} - \left(\log n + \log x\right)\right)}\\ \end{array}\]

Runtime

Time bar (total: 41.5s)Debug logProfile

herbie shell --seed 2018258 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))