Average Error: 32.8 → 16.1
Time: 2.0m
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}\;x \le 0.6908950611936806:\\ \;\;\;\;\frac{\left({\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^{3} - 1\right) \cdot \left(\left(\frac{\frac{1}{2}}{n} \cdot \log x\right) \cdot \left(\frac{\frac{1}{2}}{n} \cdot \log x\right) + \left(1 - \frac{\frac{1}{2}}{n} \cdot \log x\right)\right) - \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + 1\right)\right) \cdot \left(\frac{\log x}{n} \cdot \left(1 + {\left(\frac{\frac{1}{2}}{n} \cdot \log x\right)}^{3}\right)\right)}{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + \left(1 + {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)\right) \cdot \left(\left(1 + \frac{\frac{-1}{2}}{n} \cdot \log x\right) + \left(\frac{\frac{1}{2}}{n} \cdot \log x\right) \cdot \left(\frac{\frac{1}{2}}{n} \cdot \log x\right)\right)}\\ \mathbf{if}\;x \le 7.380653615532904 \cdot 10^{+76}:\\ \;\;\;\;\left(\frac{\frac{1}{n}}{x} + \frac{\frac{\frac{1}{2}}{n} \cdot \log x}{\frac{n}{\log x}}\right) + \frac{\frac{\frac{-1}{2}}{n} \cdot \log x}{\frac{n}{\log x}}\\ \mathbf{if}\;x \le 1.560366680492729 \cdot 10^{+95}:\\ \;\;\;\;\log \left(\sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right) + \log \left(\sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}} \cdot \sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)\\ \mathbf{if}\;x \le 1.0184148663506758 \cdot 10^{+185}:\\ \;\;\;\;\frac{\frac{\log x}{n \cdot n}}{x} + \left(\frac{\frac{-1}{2}}{x} + 1\right) \cdot \frac{1}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right) + \log \left(\sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}} \cdot \sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)\\ \end{array}\]

Error

Bits error versus x

Bits error versus n

Derivation

  1. Split input into 4 regimes

  2. if x < 0.6908950611936806

    1. Initial program 46.6

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

    2. Taylor expanded around inf 59.8

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

    3. Applied simplify15.1

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

    5. Applied flip3-+15.0

      \[\leadsto \left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - 1\right) - \color{blue}{\frac{{\left(\frac{\frac{1}{2}}{n} \cdot \log x\right)}^{3} + {1}^{3}}{\left(\frac{\frac{1}{2}}{n} \cdot \log x\right) \cdot \left(\frac{\frac{1}{2}}{n} \cdot \log x\right) + \left(1 \cdot 1 - \left(\frac{\frac{1}{2}}{n} \cdot \log x\right) \cdot 1\right)}} \cdot \frac{\log x}{n}\]

    6. Applied associate-*l/15.0

      \[\leadsto \left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - 1\right) - \color{blue}{\frac{\left({\left(\frac{\frac{1}{2}}{n} \cdot \log x\right)}^{3} + {1}^{3}\right) \cdot \frac{\log x}{n}}{\left(\frac{\frac{1}{2}}{n} \cdot \log x\right) \cdot \left(\frac{\frac{1}{2}}{n} \cdot \log x\right) + \left(1 \cdot 1 - \left(\frac{\frac{1}{2}}{n} \cdot \log x\right) \cdot 1\right)}}\]

    7. Applied flip3--15.0

      \[\leadsto \color{blue}{\frac{{\left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)}\right)}^{3} - {1}^{3}}{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(1 + x\right)}^{\left(\frac{1}{n}\right)} + \left(1 \cdot 1 + {\left(1 + x\right)}^{\left(\frac{1}{n}\right)} \cdot 1\right)}} - \frac{\left({\left(\frac{\frac{1}{2}}{n} \cdot \log x\right)}^{3} + {1}^{3}\right) \cdot \frac{\log x}{n}}{\left(\frac{\frac{1}{2}}{n} \cdot \log x\right) \cdot \left(\frac{\frac{1}{2}}{n} \cdot \log x\right) + \left(1 \cdot 1 - \left(\frac{\frac{1}{2}}{n} \cdot \log x\right) \cdot 1\right)}\]

    8. Applied frac-sub15.1

      \[\leadsto \color{blue}{\frac{\left({\left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)}\right)}^{3} - {1}^{3}\right) \cdot \left(\left(\frac{\frac{1}{2}}{n} \cdot \log x\right) \cdot \left(\frac{\frac{1}{2}}{n} \cdot \log x\right) + \left(1 \cdot 1 - \left(\frac{\frac{1}{2}}{n} \cdot \log x\right) \cdot 1\right)\right) - \left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(1 + x\right)}^{\left(\frac{1}{n}\right)} + \left(1 \cdot 1 + {\left(1 + x\right)}^{\left(\frac{1}{n}\right)} \cdot 1\right)\right) \cdot \left(\left({\left(\frac{\frac{1}{2}}{n} \cdot \log x\right)}^{3} + {1}^{3}\right) \cdot \frac{\log x}{n}\right)}{\left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(1 + x\right)}^{\left(\frac{1}{n}\right)} + \left(1 \cdot 1 + {\left(1 + x\right)}^{\left(\frac{1}{n}\right)} \cdot 1\right)\right) \cdot \left(\left(\frac{\frac{1}{2}}{n} \cdot \log x\right) \cdot \left(\frac{\frac{1}{2}}{n} \cdot \log x\right) + \left(1 \cdot 1 - \left(\frac{\frac{1}{2}}{n} \cdot \log x\right) \cdot 1\right)\right)}}\]

    9. Applied simplify15.1

      \[\leadsto \frac{\left({\left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)}\right)}^{3} - {1}^{3}\right) \cdot \left(\left(\frac{\frac{1}{2}}{n} \cdot \log x\right) \cdot \left(\frac{\frac{1}{2}}{n} \cdot \log x\right) + \left(1 \cdot 1 - \left(\frac{\frac{1}{2}}{n} \cdot \log x\right) \cdot 1\right)\right) - \left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(1 + x\right)}^{\left(\frac{1}{n}\right)} + \left(1 \cdot 1 + {\left(1 + x\right)}^{\left(\frac{1}{n}\right)} \cdot 1\right)\right) \cdot \left(\left({\left(\frac{\frac{1}{2}}{n} \cdot \log x\right)}^{3} + {1}^{3}\right) \cdot \frac{\log x}{n}\right)}{\color{blue}{\left(\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + 1\right) + {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right) \cdot \left(\left(\frac{\frac{1}{2}}{n} \cdot \log x\right) \cdot \left(\frac{\frac{1}{2}}{n} \cdot \log x\right) + \left(1 + \log x \cdot \frac{\frac{-1}{2}}{n}\right)\right)}}\]

    if 0.6908950611936806 < x < 7.380653615532904e+76

    1. Initial program 35.7

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

    2. Taylor expanded around inf 56.6

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

    3. Applied simplify61.6

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

    4. Taylor expanded around inf 49.6

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

    5. Applied simplify19.0

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

    if 7.380653615532904e+76 < x < 1.560366680492729e+95 or 1.0184148663506758e+185 < x

    1. Initial program 11.8

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

    3. Applied add-log-exp11.8

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

    4. Applied add-log-exp11.8

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

    5. Applied diff-log11.8

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

    6. Applied simplify11.8

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

    8. Applied add-cube-cbrt11.8

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

    9. Applied log-prod11.8

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

    if 1.560366680492729e+95 < x < 1.0184148663506758e+185

    1. Initial program 23.1

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

    2. Taylor expanded around inf 23.6

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

    3. Applied simplify23.6

      \[\leadsto \color{blue}{\frac{\frac{\log x}{n \cdot n}}{x} + \frac{1}{x \cdot n} \cdot \left(1 + \frac{\frac{-1}{2}}{x}\right)}\]
  3. Recombined 4 regimes into one program.

  4. Applied simplify16.1

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;x \le 0.6908950611936806:\\ \;\;\;\;\frac{\left({\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^{3} - 1\right) \cdot \left(\left(\frac{\frac{1}{2}}{n} \cdot \log x\right) \cdot \left(\frac{\frac{1}{2}}{n} \cdot \log x\right) + \left(1 - \frac{\frac{1}{2}}{n} \cdot \log x\right)\right) - \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + 1\right)\right) \cdot \left(\frac{\log x}{n} \cdot \left(1 + {\left(\frac{\frac{1}{2}}{n} \cdot \log x\right)}^{3}\right)\right)}{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + \left(1 + {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)\right) \cdot \left(\left(1 + \frac{\frac{-1}{2}}{n} \cdot \log x\right) + \left(\frac{\frac{1}{2}}{n} \cdot \log x\right) \cdot \left(\frac{\frac{1}{2}}{n} \cdot \log x\right)\right)}\\ \mathbf{if}\;x \le 7.380653615532904 \cdot 10^{+76}:\\ \;\;\;\;\left(\frac{\frac{1}{n}}{x} + \frac{\frac{\frac{1}{2}}{n} \cdot \log x}{\frac{n}{\log x}}\right) + \frac{\frac{\frac{-1}{2}}{n} \cdot \log x}{\frac{n}{\log x}}\\ \mathbf{if}\;x \le 1.560366680492729 \cdot 10^{+95}:\\ \;\;\;\;\log \left(\sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right) + \log \left(\sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}} \cdot \sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)\\ \mathbf{if}\;x \le 1.0184148663506758 \cdot 10^{+185}:\\ \;\;\;\;\frac{\frac{\log x}{n \cdot n}}{x} + \left(\frac{\frac{-1}{2}}{x} + 1\right) \cdot \frac{1}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right) + \log \left(\sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}} \cdot \sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)\\ \end{array}}\]

Runtime

Time bar (total: 2.0m)Debug logProfile

herbie shell --seed '#(1070991898 1055468627 4280279443 640792587 928206309 3646738750)' 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))