Average Error: 29.3 → 22.1
Time: 30.1s
Precision: 64
Internal Precision: 128
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -59903449304964056.0:\\ \;\;\;\;\left(1 + \frac{\frac{-1}{2}}{x}\right) \cdot \frac{\frac{1}{n}}{x} + \frac{\frac{\frac{\log x}{x}}{n}}{n}\\ \mathbf{elif}\;n \le 4123251.170831553:\\ \;\;\;\;{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - \sqrt{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt{{x}^{\left(\frac{1}{n}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{\frac{-1}{2}}{x}\right) \cdot \frac{\frac{1}{n}}{x} + \frac{\frac{\frac{\log x}{x}}{n}}{n}\\ \end{array}\]

Error

Bits error versus x

Bits error versus n

Derivation

  1. Split input into 2 regimes

  2. if n < -59903449304964056.0 or 4123251.170831553 < n

    1. Initial program 44.9

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

    3. Applied add-sqr-sqrt44.9

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

    4. Taylor expanded around inf 32.7

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

    5. Simplified32.7

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

    7. Applied associate-/r*32.1

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

    8. Taylor expanded around inf 32.1

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

    9. Simplified32.1

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

    if -59903449304964056.0 < n < 4123251.170831553

    1. Initial program 9.0

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

    3. Applied add-sqr-sqrt9.1

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

  4. Final simplification22.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -59903449304964056.0:\\ \;\;\;\;\left(1 + \frac{\frac{-1}{2}}{x}\right) \cdot \frac{\frac{1}{n}}{x} + \frac{\frac{\frac{\log x}{x}}{n}}{n}\\ \mathbf{elif}\;n \le 4123251.170831553:\\ \;\;\;\;{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - \sqrt{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt{{x}^{\left(\frac{1}{n}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{\frac{-1}{2}}{x}\right) \cdot \frac{\frac{1}{n}}{x} + \frac{\frac{\frac{\log x}{x}}{n}}{n}\\ \end{array}\]

Reproduce

herbie shell --seed 2019093 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))