Average Error: 32.5 → 23.9
Time: 36.2s
Precision: 64
Internal Precision: 1344
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -59335190.42392647 \lor \neg \left(n \le 130919932.1574956\right):\\ \;\;\;\;\left(\frac{1}{x \cdot n} + \frac{\frac{\frac{-1}{2}}{x}}{x \cdot n}\right) + \frac{\log x}{\left(x \cdot n\right) \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left({\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\left({\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \log \left(e^{{\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}}\right)\right)} \cdot \left({\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)\\ \end{array}\]

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 < -59335190.42392647 or 130919932.1574956 < n

    1. Initial program 44.5

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

    2. Initial simplification44.5

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

    3. Taylor expanded around inf 32.3

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

    4. Simplified32.2

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

    if -59335190.42392647 < n < 130919932.1574956

    1. Initial program 3.3

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

    2. Initial simplification3.3

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

    4. Applied add-sqr-sqrt3.3

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

    5. Applied add-sqr-sqrt3.3

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

    6. Applied unpow-prod-down3.3

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

    7. Applied difference-of-squares3.3

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

    9. Applied add-cbrt-cube3.3

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

    11. Applied add-log-exp3.5

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

  4. Final simplification23.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -59335190.42392647 \lor \neg \left(n \le 130919932.1574956\right):\\ \;\;\;\;\left(\frac{1}{x \cdot n} + \frac{\frac{\frac{-1}{2}}{x}}{x \cdot n}\right) + \frac{\log x}{\left(x \cdot n\right) \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left({\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\left({\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \log \left(e^{{\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}}\right)\right)} \cdot \left({\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)\\ \end{array}\]

Runtime

Time bar (total: 36.2s)Debug logProfile

herbie shell --seed 2018249 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))