Average Error: 32.5 → 23.6
Time: 40.8s
Precision: 64
Internal Precision: 1344
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -3850057.382357118 \lor \neg \left(n \le 2.791788891077261 \cdot 10^{+24}\right):\\ \;\;\;\;\frac{\frac{\frac{-1}{2}}{x}}{x \cdot n} + \left(\frac{\log x}{n \cdot \left(x \cdot n\right)} + \frac{\frac{1}{x}}{n}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(\sqrt{{x}^{\left(\frac{1}{n}\right)}} + \sqrt{e^{\frac{\log_* (1 + x)}{n}}}\right) \cdot \left(\sqrt{e^{\frac{\log_* (1 + x)}{n}}} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)} \cdot \log \left(e^{\sqrt[3]{e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}}}\right)\\ \end{array}\]

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 2 regimes

  2. if n < -3850057.382357118 or 2.791788891077261e+24 < n

    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 add-exp-log44.5

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

    5. Applied pow-exp44.5

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

    6. Simplified44.5

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

    8. Applied add-log-exp44.5

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

    9. Applied add-log-exp44.5

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

    10. Applied diff-log44.5

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

    11. Simplified44.5

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

    12. Taylor expanded around -inf 63.0

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

    13. Simplified31.7

      \[\leadsto \color{blue}{\frac{\frac{\frac{-1}{2}}{x}}{x \cdot n} + \left(\left(\frac{\frac{1}{x}}{n} + 0\right) + \frac{\log x}{n \cdot \left(x \cdot n\right)}\right)}\]

    if -3850057.382357118 < n < 2.791788891077261e+24

    1. Initial program 6.1

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

    2. Initial simplification6.1

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

    4. Applied add-exp-log6.2

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

    5. Applied pow-exp6.2

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

    6. Simplified5.3

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

    8. Applied add-log-exp5.6

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

    9. Applied add-log-exp5.7

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

    10. Applied diff-log5.7

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

    11. Simplified5.7

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

    13. Applied add-cube-cbrt5.8

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

    14. Applied exp-prod5.8

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

    15. Applied log-pow5.7

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

    17. Applied add-sqr-sqrt5.7

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

    18. Applied add-sqr-sqrt5.7

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

    19. Applied difference-of-squares5.7

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

  4. Final simplification23.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -3850057.382357118 \lor \neg \left(n \le 2.791788891077261 \cdot 10^{+24}\right):\\ \;\;\;\;\frac{\frac{\frac{-1}{2}}{x}}{x \cdot n} + \left(\frac{\log x}{n \cdot \left(x \cdot n\right)} + \frac{\frac{1}{x}}{n}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(\sqrt{{x}^{\left(\frac{1}{n}\right)}} + \sqrt{e^{\frac{\log_* (1 + x)}{n}}}\right) \cdot \left(\sqrt{e^{\frac{\log_* (1 + x)}{n}}} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)} \cdot \log \left(e^{\sqrt[3]{e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}}}\right)\\ \end{array}\]

Runtime

Time bar (total: 40.8s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes32.423.621.710.682.7%
herbie shell --seed 2018295 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))