Average Error: 32.3 → 18.4
Time: 1.3m
Precision: 64
Internal Precision: 1344
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 5.561397431847598 \cdot 10^{-251}:\\ \;\;\;\;\left(\left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - 1\right) - \frac{\log x \cdot \frac{\frac{1}{2}}{n}}{\frac{n}{\log x}}\right) - \frac{\log x}{n}\\ \mathbf{if}\;x \le 6.154225659496174 \cdot 10^{-214}:\\ \;\;\;\;\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - e^{\frac{\log x}{n}}} \cdot \sqrt[3]{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - e^{\frac{\log x}{n}}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - e^{\frac{\log x}{n}}}\right) \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - e^{\frac{\log x}{n}}}}\right) \cdot \left(\sqrt[3]{{\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} + \sqrt{e^{\frac{\log x}{n}}}} \cdot \sqrt[3]{{\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} - \sqrt{e^{\frac{\log x}{n}}}}\right)\\ \mathbf{if}\;x \le 0.6597155100183201:\\ \;\;\;\;\left(\left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - 1\right) - \frac{\log x \cdot \frac{\frac{1}{2}}{n}}{\frac{n}{\log x}}\right) - \frac{\log x}{n}\\ \mathbf{if}\;x \le 9.2034477636045 \cdot 10^{+247}:\\ \;\;\;\;\left(\frac{\frac{1}{n}}{x} - \frac{\frac{\frac{1}{2}}{n}}{x \cdot x}\right) + \frac{\log x}{\left(n \cdot x\right) \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - e^{\frac{\log x}{n}}} \cdot \sqrt[3]{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - e^{\frac{\log x}{n}}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - e^{\frac{\log x}{n}}}\right) \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - e^{\frac{\log x}{n}}}}\right) \cdot \left(\sqrt[3]{{\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} + \sqrt{e^{\frac{\log x}{n}}}} \cdot \sqrt[3]{{\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} - \sqrt{e^{\frac{\log x}{n}}}}\right)\\ \end{array}\]

Error

Bits error versus x

Bits error versus n

Derivation

  1. Split input into 3 regimes

  2. if x < 5.561397431847598e-251 or 6.154225659496174e-214 < x < 0.6597155100183201

    1. Initial program 47.5

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

    2. Taylor expanded around inf 59.9

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

    3. Applied simplify14.4

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

    if 5.561397431847598e-251 < x < 6.154225659496174e-214 or 9.2034477636045e+247 < x

    1. Initial program 17.4

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

    3. Applied pow-to-exp17.5

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

    4. Applied simplify17.4

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

    6. Applied add-cube-cbrt17.4

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

    8. Applied add-sqr-sqrt17.4

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

    9. Applied add-sqr-sqrt17.4

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

    10. Applied unpow-prod-down17.4

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

    11. Applied difference-of-squares17.5

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

    12. Applied cbrt-prod17.5

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

    14. Applied add-cube-cbrt17.5

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

    if 0.6597155100183201 < x < 9.2034477636045e+247

    1. Initial program 24.7

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

    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 simplify22.3

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

Runtime

Time bar (total: 1.3m)Debug logProfile

herbie shell --seed '#(1071725047 233389029 2036512464 3988615230 2972226563 1111574017)' 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))