Average Error: 32.4 → 22.7
Time: 2.9m
Precision: 64
Internal Precision: 1344
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\log \left(\frac{\sqrt[3]{e^{e^{\frac{\log_* (1 + x)}{n}}}} \cdot \sqrt[3]{e^{e^{\frac{\log_* (1 + x)}{n}}}}}{\sqrt{e^{{x}^{\left(\frac{1}{n}\right)}}}}\right) + \log \left(\frac{\sqrt[3]{e^{e^{\frac{\log_* (1 + x)}{n}}}}}{\sqrt{e^{{x}^{\left(\frac{1}{n}\right)}}}}\right) \le 8.569511736710562 \cdot 10^{-18}:\\ \;\;\;\;\left(\sqrt[3]{\log \left(\frac{e^{e^{\frac{\log_* (1 + x)}{n}}}}{e^{{x}^{\left(\frac{1}{n}\right)}}}\right)} \cdot \sqrt[3]{\log \left(\frac{e^{e^{\frac{\log_* (1 + x)}{n}}}}{e^{{x}^{\left(\frac{1}{n}\right)}}}\right)}\right) \cdot \sqrt[3]{\log \left(\frac{e^{e^{\frac{\log_* (1 + x)}{n}}}}{e^{{x}^{\left(\frac{1}{n}\right)}}}\right)}\\ \mathbf{elif}\;\log \left(\frac{\sqrt[3]{e^{e^{\frac{\log_* (1 + x)}{n}}}} \cdot \sqrt[3]{e^{e^{\frac{\log_* (1 + x)}{n}}}}}{\sqrt{e^{{x}^{\left(\frac{1}{n}\right)}}}}\right) + \log \left(\frac{\sqrt[3]{e^{e^{\frac{\log_* (1 + x)}{n}}}}}{\sqrt{e^{{x}^{\left(\frac{1}{n}\right)}}}}\right) \le 8.113407978602584 \cdot 10^{-14}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \left(\frac{\frac{1}{2}}{\left(x \cdot x\right) \cdot n} + \frac{-\log x}{x \cdot \left(n \cdot n\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\log \left(\frac{e^{e^{\frac{\log_* (1 + x)}{n}}}}{e^{{x}^{\left(\frac{1}{n}\right)}}}\right)} \cdot \sqrt[3]{\log \left(\frac{e^{e^{\frac{\log_* (1 + x)}{n}}}}{e^{{x}^{\left(\frac{1}{n}\right)}}}\right)}\right) \cdot \sqrt[3]{\log \left(\frac{\sqrt{e^{e^{\frac{\log_* (1 + x)}{n}}}}}{\sqrt{e^{{x}^{\left(\frac{1}{n}\right)}}}}\right) + \log \left(\frac{\sqrt{e^{e^{\frac{\log_* (1 + x)}{n}}}}}{\sqrt{e^{{x}^{\left(\frac{1}{n}\right)}}}}\right)}\\ \end{array}\]

Error

Bits error versus x

Bits error versus n

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if (+ (log (/ (* (cbrt (exp (exp (/ (log1p x) n)))) (cbrt (exp (exp (/ (log1p x) n))))) (sqrt (exp (pow x (/ 1 n)))))) (log (/ (cbrt (exp (exp (/ (log1p x) n)))) (sqrt (exp (pow x (/ 1 n))))))) < 8.569511736710562e-18

    1. Initial program 3.3

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

    3. Applied add-exp-log3.5

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

    4. Applied pow-exp3.5

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

    5. Simplified3.5

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

    7. Applied add-log-exp3.9

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

    8. Applied add-log-exp3.7

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

    9. Applied diff-log3.7

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

    11. Applied add-cube-cbrt3.7

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

    if 8.569511736710562e-18 < (+ (log (/ (* (cbrt (exp (exp (/ (log1p x) n)))) (cbrt (exp (exp (/ (log1p x) n))))) (sqrt (exp (pow x (/ 1 n)))))) (log (/ (cbrt (exp (exp (/ (log1p x) n)))) (sqrt (exp (pow x (/ 1 n))))))) < 8.113407978602584e-14

    1. Initial program 44.6

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

    2. Taylor expanded around inf 31.7

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

    3. Simplified31.1

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

    if 8.113407978602584e-14 < (+ (log (/ (* (cbrt (exp (exp (/ (log1p x) n)))) (cbrt (exp (exp (/ (log1p x) n))))) (sqrt (exp (pow x (/ 1 n)))))) (log (/ (cbrt (exp (exp (/ (log1p x) n)))) (sqrt (exp (pow x (/ 1 n)))))))

    1. Initial program 5.7

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

    3. Applied add-exp-log5.7

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

    4. Applied pow-exp5.7

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

    5. Simplified2.4

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

    7. Applied add-log-exp2.5

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

    8. Applied add-log-exp3.0

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

    9. Applied diff-log3.0

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

    11. Applied add-cube-cbrt3.2

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

    13. Applied add-sqr-sqrt3.2

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

    14. Applied add-sqr-sqrt3.1

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

    15. Applied times-frac3.2

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

    16. Applied log-prod3.2

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

  4. Final simplification22.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(\frac{\sqrt[3]{e^{e^{\frac{\log_* (1 + x)}{n}}}} \cdot \sqrt[3]{e^{e^{\frac{\log_* (1 + x)}{n}}}}}{\sqrt{e^{{x}^{\left(\frac{1}{n}\right)}}}}\right) + \log \left(\frac{\sqrt[3]{e^{e^{\frac{\log_* (1 + x)}{n}}}}}{\sqrt{e^{{x}^{\left(\frac{1}{n}\right)}}}}\right) \le 8.569511736710562 \cdot 10^{-18}:\\ \;\;\;\;\left(\sqrt[3]{\log \left(\frac{e^{e^{\frac{\log_* (1 + x)}{n}}}}{e^{{x}^{\left(\frac{1}{n}\right)}}}\right)} \cdot \sqrt[3]{\log \left(\frac{e^{e^{\frac{\log_* (1 + x)}{n}}}}{e^{{x}^{\left(\frac{1}{n}\right)}}}\right)}\right) \cdot \sqrt[3]{\log \left(\frac{e^{e^{\frac{\log_* (1 + x)}{n}}}}{e^{{x}^{\left(\frac{1}{n}\right)}}}\right)}\\ \mathbf{elif}\;\log \left(\frac{\sqrt[3]{e^{e^{\frac{\log_* (1 + x)}{n}}}} \cdot \sqrt[3]{e^{e^{\frac{\log_* (1 + x)}{n}}}}}{\sqrt{e^{{x}^{\left(\frac{1}{n}\right)}}}}\right) + \log \left(\frac{\sqrt[3]{e^{e^{\frac{\log_* (1 + x)}{n}}}}}{\sqrt{e^{{x}^{\left(\frac{1}{n}\right)}}}}\right) \le 8.113407978602584 \cdot 10^{-14}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \left(\frac{\frac{1}{2}}{\left(x \cdot x\right) \cdot n} + \frac{-\log x}{x \cdot \left(n \cdot n\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\log \left(\frac{e^{e^{\frac{\log_* (1 + x)}{n}}}}{e^{{x}^{\left(\frac{1}{n}\right)}}}\right)} \cdot \sqrt[3]{\log \left(\frac{e^{e^{\frac{\log_* (1 + x)}{n}}}}{e^{{x}^{\left(\frac{1}{n}\right)}}}\right)}\right) \cdot \sqrt[3]{\log \left(\frac{\sqrt{e^{e^{\frac{\log_* (1 + x)}{n}}}}}{\sqrt{e^{{x}^{\left(\frac{1}{n}\right)}}}}\right) + \log \left(\frac{\sqrt{e^{e^{\frac{\log_* (1 + x)}{n}}}}}{\sqrt{e^{{x}^{\left(\frac{1}{n}\right)}}}}\right)}\\ \end{array}\]

Runtime

Time bar (total: 2.9m)Debug logProfile

herbie shell --seed 2018214 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))