Average Error: 32.4 → 23.4
Time: 43.1s
Precision: 64
Internal Precision: 1344
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -2592572726271668.0 \lor \neg \left(n \le 125502170450.22118\right):\\ \;\;\;\;\frac{\frac{\frac{-1}{2}}{x}}{x \cdot n} + \left(\frac{\log x}{n \cdot \left(x \cdot n\right)} + \frac{\frac{1}{x}}{n}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt[3]{\sqrt[3]{e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}}} \cdot \sqrt[3]{\sqrt[3]{e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}}}\right) \cdot \sqrt[3]{\sqrt[3]{e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}}}\right) \cdot \left(\sqrt[3]{e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}}\right)\\ \end{array}\]

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 < -2592572726271668.0 or 125502170450.22118 < n

    1. Initial program 44.9

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

    2. Initial simplification44.9

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

    4. Applied add-exp-log44.9

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

    5. Applied pow-exp44.9

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

    6. Simplified44.9

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

    7. Taylor expanded around -inf 63.0

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

    8. Simplified32.1

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

    if -2592572726271668.0 < n < 125502170450.22118

    1. Initial program 4.6

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

    2. Initial simplification4.6

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

    4. Applied add-exp-log4.7

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

    5. Applied pow-exp4.7

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

    6. Simplified3.8

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

    8. Applied add-cube-cbrt3.8

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

    10. Applied add-cube-cbrt3.9

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

  4. Final simplification23.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -2592572726271668.0 \lor \neg \left(n \le 125502170450.22118\right):\\ \;\;\;\;\frac{\frac{\frac{-1}{2}}{x}}{x \cdot n} + \left(\frac{\log x}{n \cdot \left(x \cdot n\right)} + \frac{\frac{1}{x}}{n}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt[3]{\sqrt[3]{e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}}} \cdot \sqrt[3]{\sqrt[3]{e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}}}\right) \cdot \sqrt[3]{\sqrt[3]{e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}}}\right) \cdot \left(\sqrt[3]{e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}}\right)\\ \end{array}\]

Runtime

Time bar (total: 43.1s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes32.223.421.810.385.1%
herbie shell --seed 2018297 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))