Average Error: 32.2 → 22.3
Time: 1.4m
Precision: 64
Internal Precision: 1344
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \le 6.62614405686816 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{e^{\frac{\log x}{n}}}}{n} \cdot \frac{\sqrt{e^{\frac{\log x}{n}}}}{\frac{\frac{1}{8}}{x} + \left(\frac{1}{2} + x\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\sqrt[3]{{\left(\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right)}^{3}}}\\ \end{array}\]

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 (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))) < 6.62614405686816e-310

    1. Initial program 35.3

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

    3. Applied add-exp-log35.8

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

    4. Taylor expanded around inf 50.5

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

    5. Applied simplify24.4

      \[\leadsto \color{blue}{e^{\frac{\log x}{n}} \cdot \frac{\frac{\frac{1}{n}}{x}}{e^{\frac{\frac{1}{2}}{x}}}}\]

    6. Taylor expanded around inf 24.2

      \[\leadsto e^{\frac{\log x}{n}} \cdot \frac{\frac{\frac{1}{n}}{x}}{\color{blue}{\frac{1}{8} \cdot \frac{1}{{x}^{2}} + \left(\frac{1}{2} \cdot \frac{1}{x} + 1\right)}}\]

    7. Applied simplify24.7

      \[\leadsto \color{blue}{\frac{e^{\frac{\log x}{n}}}{n \cdot \left(\frac{\frac{1}{8}}{x} + \left(\frac{1}{2} + x\right)\right)}}\]
    8. Using strategy rm

    9. Applied add-sqr-sqrt24.7

      \[\leadsto \frac{\color{blue}{\sqrt{e^{\frac{\log x}{n}}} \cdot \sqrt{e^{\frac{\log x}{n}}}}}{n \cdot \left(\frac{\frac{1}{8}}{x} + \left(\frac{1}{2} + x\right)\right)}\]

    10. Applied times-frac24.1

      \[\leadsto \color{blue}{\frac{\sqrt{e^{\frac{\log x}{n}}}}{n} \cdot \frac{\sqrt{e^{\frac{\log x}{n}}}}{\frac{\frac{1}{8}}{x} + \left(\frac{1}{2} + x\right)}}\]

    if 6.62614405686816e-310 < (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n)))

    1. Initial program 5.8

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

    3. Applied add-exp-log5.8

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

    5. Applied add-cbrt-cube5.8

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

    6. Applied simplify5.8

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

Runtime

Time bar (total: 1.4m)Debug logProfile

herbie shell --seed 2020178 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))