Average Error: 29.0 → 18.6
Time: 5.4m
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -2531778.3601611024:\\ \;\;\;\;\left((\left(\frac{\frac{\frac{1}{4}}{n}}{n}\right) \cdot \left(\frac{\log x}{x}\right) + \left(\frac{\frac{1}{2}}{x \cdot n}\right))_* - \frac{\frac{\frac{1}{4}}{n}}{x \cdot x}\right) \cdot \left({x}^{\left(\frac{\frac{1}{n}}{2}\right)} + {\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)\\ \mathbf{elif}\;n \le -9.28993431401211 \cdot 10^{-308}:\\ \;\;\;\;(\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right) + \left(-{x}^{\left(\frac{1}{n}\right)}\right))_*\\ \mathbf{elif}\;n \le 1488923677953.9617:\\ \;\;\;\;e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{\frac{-1}{2}}{x}\right) \cdot \left(\frac{1}{x \cdot n}\right) + \left(\frac{1}{x \cdot n} - \frac{-\log x}{n \cdot \left(x \cdot n\right)}\right))_*\\ \end{array}\]

Error

Bits error versus x

Bits error versus n

Derivation

  1. Split input into 4 regimes

  2. if n < -2531778.3601611024

    1. Initial program 45.0

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

    3. Applied sqr-pow45.1

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

    4. Applied sqr-pow45.1

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

    5. Applied difference-of-squares45.1

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

    6. Taylor expanded around inf 31.9

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

    7. Simplified31.9

      \[\leadsto \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \color{blue}{\left((\left(\frac{\frac{\frac{1}{4}}{n}}{n}\right) \cdot \left(\frac{\log x}{x}\right) + \left(\frac{\frac{1}{2}}{x \cdot n}\right))_* - \frac{\frac{\frac{1}{4}}{n}}{x \cdot x}\right)}\]

    if -2531778.3601611024 < n < -9.28993431401211e-308

    1. Initial program 0.6

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

    3. Applied add-sqr-sqrt0.6

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

    4. Applied fma-neg0.6

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

    if -9.28993431401211e-308 < n < 1488923677953.9617

    1. Initial program 23.3

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

    3. Applied add-exp-log23.4

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

    4. Applied pow-exp23.4

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

    5. Simplified2.4

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

    if 1488923677953.9617 < 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-log-exp44.9

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

    4. Taylor expanded around inf 31.3

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

    5. Simplified31.3

      \[\leadsto \color{blue}{(\left(\frac{\frac{-1}{2}}{x}\right) \cdot \left(\frac{1}{n \cdot x}\right) + \left(\frac{1}{n \cdot x} - \frac{-\log x}{n \cdot \left(n \cdot x\right)}\right))_*}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification18.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -2531778.3601611024:\\ \;\;\;\;\left((\left(\frac{\frac{\frac{1}{4}}{n}}{n}\right) \cdot \left(\frac{\log x}{x}\right) + \left(\frac{\frac{1}{2}}{x \cdot n}\right))_* - \frac{\frac{\frac{1}{4}}{n}}{x \cdot x}\right) \cdot \left({x}^{\left(\frac{\frac{1}{n}}{2}\right)} + {\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)\\ \mathbf{elif}\;n \le -9.28993431401211 \cdot 10^{-308}:\\ \;\;\;\;(\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right) + \left(-{x}^{\left(\frac{1}{n}\right)}\right))_*\\ \mathbf{elif}\;n \le 1488923677953.9617:\\ \;\;\;\;e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{\frac{-1}{2}}{x}\right) \cdot \left(\frac{1}{x \cdot n}\right) + \left(\frac{1}{x \cdot n} - \frac{-\log x}{n \cdot \left(x \cdot n\right)}\right))_*\\ \end{array}\]

Reproduce

herbie shell --seed 2019100 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))