Average Error: 32.6 → 22.7
Time: 1.7m
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}\;\frac{1}{n} \le -5.363094132516128 \cdot 10^{-12}:\\ \;\;\;\;\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({\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\left({\left(\sqrt{\sqrt{x + 1}}\right)}^{\left(\frac{1}{n}\right)} + \sqrt{\sqrt{{x}^{\left(\frac{1}{n}\right)}}}\right) \cdot \log \left(e^{{\left(\sqrt{\sqrt{1 + x}}\right)}^{\left(\frac{1}{n}\right)} - \sqrt{\sqrt{{x}^{\left(\frac{1}{n}\right)}}}}\right)\right)}\\ \mathbf{if}\;\frac{1}{n} \le -1.4555907476059826 \cdot 10^{-116}:\\ \;\;\;\;\left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - 1\right) - (\left(\frac{\log x}{n}\right) \cdot \left(\frac{\frac{1}{2}}{n} \cdot \log x\right) + \left(\frac{\log x}{n}\right))_*\\ \mathbf{if}\;\frac{1}{n} \le -1.3667660918061747 \cdot 10^{-258}:\\ \;\;\;\;\left(\frac{\frac{1}{n}}{x} - \frac{\frac{\frac{1}{2}}{x}}{n \cdot x}\right) + \frac{\frac{\log x}{n}}{n \cdot x}\\ \mathbf{if}\;\frac{1}{n} \le 3.711468536197641 \cdot 10^{-292}:\\ \;\;\;\;\left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - 1\right) - (\left(\frac{\log x}{n}\right) \cdot \left(\frac{\frac{1}{2}}{n} \cdot \log x\right) + \left(\frac{\log x}{n}\right))_*\\ \mathbf{if}\;\frac{1}{n} \le 1.7716243371803304 \cdot 10^{-266}:\\ \;\;\;\;\left(\frac{\frac{1}{n}}{x} - \frac{\frac{\frac{1}{2}}{x}}{n \cdot x}\right) + \frac{\frac{\log x}{n}}{n \cdot x}\\ \mathbf{if}\;\frac{1}{n} \le 3.5335224200538646 \cdot 10^{-260}:\\ \;\;\;\;\left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - 1\right) - (\left(\frac{\log x}{n}\right) \cdot \left(\frac{\frac{1}{2}}{n} \cdot \log x\right) + \left(\frac{\log x}{n}\right))_*\\ \mathbf{if}\;\frac{1}{n} \le 5.85591473108959 \cdot 10^{-110}:\\ \;\;\;\;\left(\frac{\frac{1}{n}}{x} - \frac{\frac{\frac{1}{2}}{x}}{n \cdot x}\right) + \frac{\frac{\log x}{n}}{n \cdot x}\\ \mathbf{if}\;\frac{1}{n} \le 3.594664189067294 \cdot 10^{-15}:\\ \;\;\;\;\left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - 1\right) - (\left(\frac{\log x}{n}\right) \cdot \left(\frac{\frac{1}{2}}{n} \cdot \log x\right) + \left(\frac{\log x}{n}\right))_*\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array}\]

Error

Bits error versus x

Bits error versus n

Derivation

  1. Split input into 4 regimes

  2. if (/ 1 n) < -5.363094132516128e-12

    1. Initial program 1.9

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

    3. Applied add-cube-cbrt1.9

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

    5. Applied add-sqr-sqrt1.9

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

    6. Applied add-sqr-sqrt1.9

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

    7. Applied unpow-prod-down1.9

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

    8. Applied difference-of-squares1.9

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

    10. Applied add-sqr-sqrt1.9

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

    11. Applied sqrt-prod2.0

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

    12. Applied add-sqr-sqrt2.0

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

    13. Applied sqrt-prod2.0

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

    14. Applied unpow-prod-down2.0

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

    15. Applied difference-of-squares1.9

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

    17. Applied add-log-exp2.1

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

    18. Applied add-log-exp2.1

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

    19. Applied diff-log2.1

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

    20. Applied simplify2.1

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

    if -5.363094132516128e-12 < (/ 1 n) < -1.4555907476059826e-116 or -1.3667660918061747e-258 < (/ 1 n) < 3.711468536197641e-292 or 1.7716243371803304e-266 < (/ 1 n) < 3.5335224200538646e-260 or 5.85591473108959e-110 < (/ 1 n) < 3.594664189067294e-15

    1. Initial program 49.0

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

    2. Taylor expanded around inf 49.0

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

    3. Applied simplify30.9

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

    if -1.4555907476059826e-116 < (/ 1 n) < -1.3667660918061747e-258 or 3.711468536197641e-292 < (/ 1 n) < 1.7716243371803304e-266 or 3.5335224200538646e-260 < (/ 1 n) < 5.85591473108959e-110

    1. Initial program 42.2

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

    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 simplify31.8

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

    if 3.594664189067294e-15 < (/ 1 n)

    1. Initial program 8.3

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

    3. Applied add-exp-log8.3

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

    4. Applied pow-exp8.3

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

    5. Applied simplify5.4

      \[\leadsto e^{\color{blue}{\frac{\log_* (1 + x)}{n}}} - {x}^{\left(\frac{1}{n}\right)}\]
  3. Recombined 4 regimes into one program.

Runtime

Time bar (total: 1.7m)Debug logProfile

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