Average Error: 32.8 → 24.6
Time: 1.5m
Precision: 64
Internal Precision: 1344
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -0.08886812668802994:\\ \;\;\;\;\sqrt[3]{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - e^{\frac{1}{n} \cdot \log x}\right) \cdot \left(\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - e^{\frac{1}{n} \cdot \log x}\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - e^{\frac{1}{n} \cdot \log x}\right)\right)}\\ \mathbf{elif}\;\frac{1}{n} \le 1.095760740977552 \cdot 10^{-15}:\\ \;\;\;\;\frac{\log x}{n \cdot \left(x \cdot n\right)} + \left(\frac{1}{x \cdot n} + \frac{\frac{\frac{-1}{2}}{x}}{x \cdot n}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\left(\left(\left({x}^{\left(\frac{1}{n}\right)} + {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right) \cdot \left(\left({x}^{\left(\frac{1}{n}\right)} + {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right)\right) \cdot \left({\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^{3} - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^{3}\right)}}{\sqrt[3]{\left(\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + e^{\frac{1}{n} \cdot \log x}\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + e^{\frac{1}{n} \cdot \log x}\right)\right) \cdot \left(\left(e^{\frac{1}{n} \cdot \log x} \cdot {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + e^{\frac{1}{n} \cdot \log x} \cdot e^{\frac{1}{n} \cdot \log x}\right) + {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}}\\ \end{array}\]

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) < -0.08886812668802994

    1. Initial program 0.2

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

    2. Initial simplification0.2

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

    4. Applied pow-to-exp0.2

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

    6. Applied add-cbrt-cube0.3

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

    8. Applied *-un-lft-identity0.3

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

    if -0.08886812668802994 < (/ 1 n) < 1.095760740977552e-15

    1. Initial program 44.5

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

    2. Initial simplification44.5

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

    4. Applied pow-to-exp44.5

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

    5. Taylor expanded around inf 33.0

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

    6. 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 1.095760740977552e-15 < (/ 1 n)

    1. Initial program 10.3

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

    2. Initial simplification10.3

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

    4. Applied pow-to-exp10.3

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

    6. Applied add-cbrt-cube10.4

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

    8. Applied flip3--10.4

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

    9. Applied flip--10.4

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

    10. Applied flip--10.4

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

    11. Applied frac-times10.4

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

    12. Applied frac-times10.4

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

    13. Applied cbrt-div10.4

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

    14. Simplified10.4

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

  4. Final simplification24.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -0.08886812668802994:\\ \;\;\;\;\sqrt[3]{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - e^{\frac{1}{n} \cdot \log x}\right) \cdot \left(\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - e^{\frac{1}{n} \cdot \log x}\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - e^{\frac{1}{n} \cdot \log x}\right)\right)}\\ \mathbf{elif}\;\frac{1}{n} \le 1.095760740977552 \cdot 10^{-15}:\\ \;\;\;\;\frac{\log x}{n \cdot \left(x \cdot n\right)} + \left(\frac{1}{x \cdot n} + \frac{\frac{\frac{-1}{2}}{x}}{x \cdot n}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\left(\left(\left({x}^{\left(\frac{1}{n}\right)} + {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right) \cdot \left(\left({x}^{\left(\frac{1}{n}\right)} + {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right)\right) \cdot \left({\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^{3} - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^{3}\right)}}{\sqrt[3]{\left(\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + e^{\frac{1}{n} \cdot \log x}\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + e^{\frac{1}{n} \cdot \log x}\right)\right) \cdot \left(\left(e^{\frac{1}{n} \cdot \log x} \cdot {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + e^{\frac{1}{n} \cdot \log x} \cdot e^{\frac{1}{n} \cdot \log x}\right) + {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}}\\ \end{array}\]

Runtime

Time bar (total: 1.5m)Debug logProfile

herbie shell --seed 2018256 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))