Average Error: 32.6 → 23.3
Time: 48.5s
Precision: 64
Internal Precision: 1344
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -241758223396.2354 \lor \neg \left(n \le 10133725612152484.0\right):\\ \;\;\;\;\frac{\frac{\frac{-1}{2}}{x}}{x \cdot n} + \left(\frac{\log x}{n \cdot \left(x \cdot n\right)} + \frac{\frac{1}{x}}{n}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{e^{\frac{\log_* (1 + x)}{n}} - (e^{\log_* (1 + {x}^{\left(\frac{1}{n}\right)})} - 1)^*} \cdot \sqrt[3]{e^{\frac{\log_* (1 + x)}{n}} - (e^{\log_* (1 + {x}^{\left(\frac{1}{n}\right)})} - 1)^*}\right) \cdot \log \left(e^{\sqrt[3]{e^{\frac{\log_* (1 + x)}{n}} - (e^{\log_* (1 + {x}^{\left(\frac{1}{n}\right)})} - 1)^*}}\right)\\ \end{array}\]

Error

Bits error versus x

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if n < -241758223396.2354 or 10133725612152484.0 < n

    1. Initial program 44.5

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

    3. Applied add-exp-log44.5

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

    4. Applied pow-exp44.5

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

    5. Simplified44.5

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

    6. Taylor expanded around -inf 63.0

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

    7. Simplified31.4

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

    if -241758223396.2354 < n < 10133725612152484.0

    1. Initial program 5.2

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

    3. Applied add-exp-log5.2

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

    4. Applied pow-exp5.2

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

    5. Simplified4.2

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

    7. Applied expm1-log1p-u4.2

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

    9. Applied add-cube-cbrt4.3

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

    11. Applied add-log-exp4.6

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

  4. Final simplification23.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -241758223396.2354 \lor \neg \left(n \le 10133725612152484.0\right):\\ \;\;\;\;\frac{\frac{\frac{-1}{2}}{x}}{x \cdot n} + \left(\frac{\log x}{n \cdot \left(x \cdot n\right)} + \frac{\frac{1}{x}}{n}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{e^{\frac{\log_* (1 + x)}{n}} - (e^{\log_* (1 + {x}^{\left(\frac{1}{n}\right)})} - 1)^*} \cdot \sqrt[3]{e^{\frac{\log_* (1 + x)}{n}} - (e^{\log_* (1 + {x}^{\left(\frac{1}{n}\right)})} - 1)^*}\right) \cdot \log \left(e^{\sqrt[3]{e^{\frac{\log_* (1 + x)}{n}} - (e^{\log_* (1 + {x}^{\left(\frac{1}{n}\right)})} - 1)^*}}\right)\\ \end{array}\]

Runtime

Time bar (total: 48.5s)Debug logProfile

herbie shell --seed 2018248 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))