Average Error: 29.3 → 21.5
Time: 39.3s
Precision: 64
Internal Precision: 128
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -24.184674423494144:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \le 1.253338532993033 \cdot 10^{-30}:\\ \;\;\;\;\frac{\frac{-1}{x \cdot n}}{\frac{\frac{-1}{2}}{x} - 1} + \frac{\log x}{n \cdot \left(x \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array}\]

Error

Bits error versus x

Bits error versus n

Derivation

  1. Split input into 2 regimes

  2. if (/ 1 n) < -24.184674423494144 or 1.253338532993033e-30 < (/ 1 n)

    1. Initial program 9.8

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

    if -24.184674423494144 < (/ 1 n) < 1.253338532993033e-30

    1. Initial program 44.4

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

    2. Taylor expanded around inf 33.1

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

    3. Simplified33.0

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

    5. Applied associate-/r*32.4

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

    7. Applied flip-+32.4

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

    8. Applied associate-*l/32.4

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

    9. Simplified32.5

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

    10. Taylor expanded around -inf 30.5

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

    11. Simplified30.5

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

  4. Final simplification21.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -24.184674423494144:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;\frac{1}{n} \le 1.253338532993033 \cdot 10^{-30}:\\ \;\;\;\;\frac{\frac{-1}{x \cdot n}}{\frac{\frac{-1}{2}}{x} - 1} + \frac{\log x}{n \cdot \left(x \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019053 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))