Average Error: 33.1 → 23.4
Time: 55.8s
Precision: 64
Internal Precision: 1344
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\left(\frac{\log x}{n} + 1\right) + \frac{\frac{1}{n}}{x}\right) - {x}^{\left(\frac{1}{n}\right)} \le -0.010295450963190445:\\ \;\;\;\;\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\\ \mathbf{elif}\;\left(\left(\frac{\log x}{n} + 1\right) + \frac{\frac{1}{n}}{x}\right) - {x}^{\left(\frac{1}{n}\right)} \le 5.206381854628619 \cdot 10^{-296}:\\ \;\;\;\;\frac{1}{x \cdot n} + \left(\left(-\frac{\frac{\frac{1}{2}}{n}}{x \cdot x}\right) + \frac{\frac{\log x}{n}}{x \cdot n}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)} \cdot \sqrt[3]{e^{\log \left(\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\right)}}\right) \cdot \sqrt[3]{\log \left(e^{\sqrt{{x}^{\left(\frac{1}{n}\right)}} + {\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)}}\right) \cdot \left({\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)}\\ \end{array}\]

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 (- (+ (+ (/ (log x) n) 1) (/ (/ 1 n) x)) (pow x (/ 1 n))) < -0.010295450963190445

    1. Initial program 19.8

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

    3. Applied add-log-exp20.2

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

    4. Applied add-log-exp20.1

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

    5. Applied diff-log20.1

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

    6. Simplified20.1

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

    if -0.010295450963190445 < (- (+ (+ (/ (log x) n) 1) (/ (/ 1 n) x)) (pow x (/ 1 n))) < 5.206381854628619e-296

    1. Initial program 40.7

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

    2. Taylor expanded around -inf 63.0

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

    3. Simplified22.3

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

    if 5.206381854628619e-296 < (- (+ (+ (/ (log x) n) 1) (/ (/ 1 n) x)) (pow x (/ 1 n)))

    1. Initial program 31.2

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

    3. Applied add-log-exp31.3

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

    4. Applied add-log-exp31.3

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

    5. Applied diff-log31.3

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

    6. Simplified31.3

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

    8. Applied add-cube-cbrt31.3

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

    10. Applied add-sqr-sqrt31.3

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

    11. Applied add-sqr-sqrt31.3

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

    12. Applied unpow-prod-down31.3

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

    13. Applied difference-of-squares31.3

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

    14. Applied exp-prod31.3

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

    15. Applied log-pow31.3

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

    17. Applied add-exp-log31.3

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

  4. Final simplification23.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\frac{\log x}{n} + 1\right) + \frac{\frac{1}{n}}{x}\right) - {x}^{\left(\frac{1}{n}\right)} \le -0.010295450963190445:\\ \;\;\;\;\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\\ \mathbf{elif}\;\left(\left(\frac{\log x}{n} + 1\right) + \frac{\frac{1}{n}}{x}\right) - {x}^{\left(\frac{1}{n}\right)} \le 5.206381854628619 \cdot 10^{-296}:\\ \;\;\;\;\frac{1}{x \cdot n} + \left(\left(-\frac{\frac{\frac{1}{2}}{n}}{x \cdot x}\right) + \frac{\frac{\log x}{n}}{x \cdot n}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)} \cdot \sqrt[3]{e^{\log \left(\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\right)}}\right) \cdot \sqrt[3]{\log \left(e^{\sqrt{{x}^{\left(\frac{1}{n}\right)}} + {\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)}}\right) \cdot \left({\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)}\\ \end{array}\]

Runtime

Time bar (total: 55.8s)Debug logProfile

herbie shell --seed 2018217 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))