Average Error: 32.6 → 22.5
Time: 1.0m
Precision: 64
Internal Precision: 1408
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\frac{\frac{1}{n}}{x} + \frac{\log x}{n}\right) + \left(1 - {x}^{\left(\frac{1}{n}\right)}\right) \le -0.11042480760696727:\\ \;\;\;\;\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\sqrt[3]{{\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}} \cdot \left(\log \left(\sqrt[3]{e^{\sqrt[3]{{\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}}}} \cdot \sqrt[3]{e^{\sqrt[3]{{\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}}}}\right) + \log \left(\sqrt[3]{e^{\sqrt[3]{{\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}}}}\right)\right)\right)\\ \mathbf{if}\;\left(\frac{\frac{1}{n}}{x} + \frac{\log x}{n}\right) + \left(1 - {x}^{\left(\frac{1}{n}\right)}\right) \le 5.602426177808234 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{\log x}{n}}{n \cdot x} + \left(\frac{\frac{1}{n}}{x} - \frac{\frac{\frac{1}{2}}{x}}{n \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{\log \left(e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}\right) \cdot \left(\sqrt[3]{{\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}} \cdot \log \left(e^{\sqrt[3]{{\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}}}\right)\right)\\ \end{array}\]

Error

Bits error versus x

Bits error versus n

Derivation

  1. Split input into 3 regimes

  2. if (+ (+ (/ (/ 1 n) x) (/ (log x) n)) (- 1 (pow x (/ 1 n)))) < -0.11042480760696727

    1. Initial program 18.9

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

    3. Applied add-cube-cbrt18.9

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

    5. Applied add-sqr-sqrt18.9

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

    6. Applied add-sqr-sqrt18.9

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

    7. Applied unpow-prod-down18.9

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

    8. Applied difference-of-squares18.9

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

    9. Applied cbrt-prod18.9

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

    11. Applied add-log-exp19.0

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

    13. Applied add-cube-cbrt19.0

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

    14. Applied log-prod19.0

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

    if -0.11042480760696727 < (+ (+ (/ (/ 1 n) x) (/ (log x) n)) (- 1 (pow x (/ 1 n)))) < 5.602426177808234e-11

    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

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

    3. Applied simplify21.4

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

    if 5.602426177808234e-11 < (+ (+ (/ (/ 1 n) x) (/ (log x) n)) (- 1 (pow x (/ 1 n))))

    1. Initial program 30.9

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

    3. Applied add-cube-cbrt30.9

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

    5. Applied add-sqr-sqrt30.9

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

    6. Applied add-sqr-sqrt30.9

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

    7. Applied unpow-prod-down30.9

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

    8. Applied difference-of-squares30.9

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

    9. Applied cbrt-prod30.9

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

    11. Applied add-log-exp30.9

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

    13. Applied add-log-exp30.9

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

    14. Applied add-log-exp30.9

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

    15. Applied diff-log30.9

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

    16. Applied simplify30.9

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

Runtime

Time bar (total: 1.0m)Debug logProfile

herbie shell --seed '#(1070355188 2193211668 3977393919 3454156579 3755371326 1656365382)' 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))