Average Error: 29.7 → 22.7
Time: 33.1s
Precision: 64
Internal Precision: 128
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -72.46454970373382:\\ \;\;\;\;\log \left(\sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}} \cdot \sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right) + \log \left(\sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)\\ \mathbf{elif}\;\frac{1}{n} \le 5.813641785194493 \cdot 10^{-07}:\\ \;\;\;\;\left(\frac{1}{x \cdot n} + \frac{\frac{\frac{-1}{2}}{x}}{x \cdot n}\right) + \frac{\log x}{\left(x \cdot n\right) \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\log \left(\sqrt{e^{-{x}^{\left(\frac{1}{n}\right)}}}\right) + \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + \log \left(\sqrt{e^{-{x}^{\left(\frac{1}{n}\right)}}}\right)\right)} \cdot \left(\sqrt[3]{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)} \cdot \sqrt[3]{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}\right)\\ \end{array}\]

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 (/ 1 n) < -72.46454970373382

    1. Initial program 0

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

    2. Initial simplification0

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

    4. Applied add-log-exp0

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

    6. Applied add-cube-cbrt0

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

    7. Applied log-prod0

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

    if -72.46454970373382 < (/ 1 n) < 5.813641785194493e-07

    1. Initial program 44.9

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

    2. Initial simplification44.9

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

    3. Taylor expanded around inf 33.0

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

    4. Simplified33.0

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

    if 5.813641785194493e-07 < (/ 1 n)

    1. Initial program 25.3

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

    2. Initial simplification25.3

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

    4. Applied add-log-exp25.3

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

    6. Applied add-cube-cbrt25.3

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

    8. Applied sub-neg25.3

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

    9. Applied exp-sum25.3

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

    10. Applied log-prod25.3

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

    11. Simplified25.3

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

    13. Applied add-sqr-sqrt25.3

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

    14. Applied log-prod25.3

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

    15. Applied associate-+r+25.3

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

  4. Final simplification22.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -72.46454970373382:\\ \;\;\;\;\log \left(\sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}} \cdot \sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right) + \log \left(\sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)\\ \mathbf{elif}\;\frac{1}{n} \le 5.813641785194493 \cdot 10^{-07}:\\ \;\;\;\;\left(\frac{1}{x \cdot n} + \frac{\frac{\frac{-1}{2}}{x}}{x \cdot n}\right) + \frac{\log x}{\left(x \cdot n\right) \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\log \left(\sqrt{e^{-{x}^{\left(\frac{1}{n}\right)}}}\right) + \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + \log \left(\sqrt{e^{-{x}^{\left(\frac{1}{n}\right)}}}\right)\right)} \cdot \left(\sqrt[3]{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)} \cdot \sqrt[3]{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}\right)\\ \end{array}\]

Runtime

Time bar (total: 33.1s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes29.822.721.38.484.1%
herbie shell --seed 2018352 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))