Average Error: 33.0 → 23.9
Time: 16.9s
Precision: binary64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;n \le -85.391473878602653:\\ \;\;\;\;\left(\frac{\frac{1}{n}}{x} + 1 \cdot \left(0 - \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}\right)\right) - \log \left(e^{\frac{\frac{0.5}{n}}{{x}^{2}}}\right)\\ \mathbf{elif}\;n \le 8033134678603.64355:\\ \;\;\;\;e^{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\left(\left(\left(-\left(\log n + \log x\right)\right) + 1 \cdot \frac{\log 1}{n}\right) + \log 1\right) - 0.5 \cdot \frac{1}{x}\right) - 1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}\\ \end{array}\]

Error

Bits error versus x

Bits error versus n

Derivation

  1. Split input into 3 regimes

  2. if n < -85.391473878602653

    1. Initial program 44.3

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

    2. Taylor expanded around inf 32.5

      \[\leadsto \color{blue}{\left(1 \cdot \frac{1}{x \cdot n} + 1 \cdot \frac{\log 1}{x \cdot {n}^{2}}\right) - \left(0.5 \cdot \frac{1}{{x}^{2} \cdot n} + 1 \cdot \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}\right)}\]

    3. Simplified31.7

      \[\leadsto \color{blue}{\left(\frac{\frac{1}{n}}{x} + 1 \cdot \left(0 - \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}\right)\right) - \frac{\frac{0.5}{n}}{{x}^{2}}}\]
    4. Using strategy rm

    5. Applied add-log-exp31.8

      \[\leadsto \left(\frac{\frac{1}{n}}{x} + 1 \cdot \left(0 - \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}\right)\right) - \color{blue}{\log \left(e^{\frac{\frac{0.5}{n}}{{x}^{2}}}\right)}\]

    if -85.391473878602653 < n < 8033134678603.64355

    1. Initial program 3.6

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

    3. Applied add-exp-log4.0

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

    if 8033134678603.64355 < n

    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 add-exp-log45.0

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

    4. Taylor expanded around inf 32.0

      \[\leadsto e^{\color{blue}{\left(\log \left(\frac{1}{n}\right) + \left(\log \left(\frac{1}{x}\right) + \left(\log 1 + 1 \cdot \frac{\log 1}{n}\right)\right)\right) - \left(1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n} + 0.5 \cdot \frac{1}{x}\right)}}\]

    5. Simplified32.0

      \[\leadsto e^{\color{blue}{\left(\left(\left(\left(-\left(\log n + \log x\right)\right) + 1 \cdot \frac{\log 1}{n}\right) + \log 1\right) - 0.5 \cdot \frac{1}{x}\right) - 1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification23.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -85.391473878602653:\\ \;\;\;\;\left(\frac{\frac{1}{n}}{x} + 1 \cdot \left(0 - \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}\right)\right) - \log \left(e^{\frac{\frac{0.5}{n}}{{x}^{2}}}\right)\\ \mathbf{elif}\;n \le 8033134678603.64355:\\ \;\;\;\;e^{\log \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\left(\left(\left(-\left(\log n + \log x\right)\right) + 1 \cdot \frac{\log 1}{n}\right) + \log 1\right) - 0.5 \cdot \frac{1}{x}\right) - 1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020147 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))