Average Error: 29.0 → 22.0
Time: 41.2s
Precision: 64
Internal Precision: 128
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -4.865667609282194 \cdot 10^{-16}:\\ \;\;\;\;\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \left(\log \left(\sqrt[3]{e^{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}} \cdot \sqrt[3]{e^{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}}\right) + \log \left(\sqrt[3]{e^{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}}\right)\right)\\ \mathbf{elif}\;\frac{1}{n} \le 4.084258086542354 \cdot 10^{-08}:\\ \;\;\;\;\left(\frac{\frac{\frac{-1}{2}}{x}}{x \cdot n} + \frac{1}{x \cdot n}\right) + \frac{\log x}{n \cdot \left(x \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right) \cdot \sqrt[3]{e^{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}}\\ \end{array}\]

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) < -4.865667609282194e-16

    1. Initial program 1.5

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

    2. Initial simplification1.5

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

    4. Applied add-log-exp1.7

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

    5. Applied add-log-exp1.6

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

    6. Applied diff-log1.6

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

    7. Simplified1.6

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

    9. Applied add-cube-cbrt1.6

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

    10. Applied exp-prod1.6

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

    11. Applied log-pow1.6

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

    13. Applied add-cube-cbrt1.6

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

    14. Applied log-prod1.6

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

    if -4.865667609282194e-16 < (/ 1 n) < 4.084258086542354e-08

    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-log-exp45.3

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

    5. Applied add-log-exp45.3

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

    6. Applied diff-log45.3

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

    7. Simplified45.3

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

    8. Taylor expanded around inf 32.6

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

    9. Simplified32.5

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

    if 4.084258086542354e-08 < (/ 1 n)

    1. Initial program 24.9

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

    2. Initial simplification24.9

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

    4. Applied add-log-exp24.9

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

    5. Applied add-log-exp25.0

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

    6. Applied diff-log25.0

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

    7. Simplified25.0

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

    9. Applied add-cube-cbrt25.0

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

    10. Applied exp-prod25.0

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

    11. Applied log-pow24.9

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

    13. Applied add-exp-log24.9

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

  4. Final simplification22.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -4.865667609282194 \cdot 10^{-16}:\\ \;\;\;\;\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \left(\log \left(\sqrt[3]{e^{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}} \cdot \sqrt[3]{e^{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}}\right) + \log \left(\sqrt[3]{e^{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}}\right)\right)\\ \mathbf{elif}\;\frac{1}{n} \le 4.084258086542354 \cdot 10^{-08}:\\ \;\;\;\;\left(\frac{\frac{\frac{-1}{2}}{x}}{x \cdot n} + \frac{1}{x \cdot n}\right) + \frac{\log x}{n \cdot \left(x \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right) \cdot \sqrt[3]{e^{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}}\\ \end{array}\]

Runtime

Time bar (total: 41.2s)Debug logProfile

herbie shell --seed 2018296 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))