Average Error: 29.4 → 21.6
Time: 37.7s
Precision: 64
Internal Precision: 128
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -1082.3055037159368:\\ \;\;\;\;\frac{\log x}{\left(x \cdot n\right) \cdot n} + \left(\frac{\frac{1}{n}}{x} + \log \left(e^{\frac{\frac{\frac{-1}{2}}{x}}{x \cdot n}}\right)\right)\\ \mathbf{elif}\;n \le 299549.017404473:\\ \;\;\;\;\sqrt[3]{\left(\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{\frac{-1}{2}}{x}}{x \cdot n} + \frac{1}{n} \cdot \frac{1}{x}\right) + \frac{\log x}{\left(x \cdot n\right) \cdot n}\\ \end{array}\]

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 3 regimes

  2. if n < -1082.3055037159368

    1. Initial program 45.1

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

    2. Taylor expanded around inf 32.3

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

    3. Simplified32.3

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

    5. Applied associate-/r*31.7

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

    7. Applied add-log-exp31.8

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

    if -1082.3055037159368 < n < 299549.017404473

    1. Initial program 7.5

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

    3. Applied add-cbrt-cube7.5

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

    if 299549.017404473 < n

    1. Initial program 46.0

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

    2. Taylor expanded around inf 32.9

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

    3. Simplified32.8

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

    5. Applied associate-/r*32.3

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

    7. Applied div-inv32.3

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

  4. Final simplification21.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -1082.3055037159368:\\ \;\;\;\;\frac{\log x}{\left(x \cdot n\right) \cdot n} + \left(\frac{\frac{1}{n}}{x} + \log \left(e^{\frac{\frac{\frac{-1}{2}}{x}}{x \cdot n}}\right)\right)\\ \mathbf{elif}\;n \le 299549.017404473:\\ \;\;\;\;\sqrt[3]{\left(\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{\frac{-1}{2}}{x}}{x \cdot n} + \frac{1}{n} \cdot \frac{1}{x}\right) + \frac{\log x}{\left(x \cdot n\right) \cdot n}\\ \end{array}\]

Runtime

Time bar (total: 37.7s)Debug logProfile

herbie shell --seed 2018278 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))