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

⬇

\[\begin{array}{l} \mathbf{if}\;n \le -31.329673016395393:\\ \;\;\;\;\left(\frac{\frac{1}{n}}{x} - \frac{\frac{\frac{1}{2}}{x}}{n \cdot x}\right) - \frac{\frac{\log x}{n \cdot n}}{x}\\ \mathbf{if}\;n \le 5667669302503732.0:\\ \;\;\;\;{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}^3\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{1}{n}}{x} - \frac{\frac{\frac{1}{2}}{x}}{n \cdot x}\right) - \frac{\frac{\log x}{n \cdot n}}{x}\\ \end{array}\]

Derivation

Split input into 2 regimes.
if n < -31.329673016395393 or 5667669302503732.0 < n
1. Initial program 44.0
  \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
2. Applied taylor 8.9
  \[\leadsto \frac{1}{x \cdot n} - \left(\frac{1}{2} \cdot \frac{1}{{x}^2 \cdot n} + \frac{\log x}{x \cdot {n}^2}\right)\]
3. Taylor expanded around inf 8.9
  
  \[\leadsto \color{blue}{\frac{1}{x \cdot n} - \left(\frac{1}{2} \cdot \frac{1}{{x}^2 \cdot n} + \frac{\log x}{x \cdot {n}^2}\right)}\]
4. Applied simplify 8.9
  \[\leadsto \color{blue}{\left(\frac{1}{n \cdot x} - \frac{\frac{\frac{1}{2}}{n}}{{x}^2}\right) - \frac{\log x}{n \cdot \left(n \cdot x\right)}}\]
5. Applied taylor 8.9
  \[\leadsto \left(\frac{1}{x \cdot n} - \frac{1}{2} \cdot \frac{1}{{x}^2 \cdot n}\right) - \frac{\log x}{n \cdot \left(n \cdot x\right)}\]
6. Taylor expanded around 0 8.9
  
  \[\leadsto \color{blue}{\left(\frac{1}{x \cdot n} - \frac{1}{2} \cdot \frac{1}{{x}^2 \cdot n}\right)} - \frac{\log x}{n \cdot \left(n \cdot x\right)}\]
7. Applied simplify 0.3
  \[\leadsto \color{blue}{\left(\frac{\frac{1}{n}}{x} - \frac{\frac{\frac{1}{2}}{x}}{n \cdot x}\right) - \frac{\frac{\log x}{n \cdot n}}{x}}\]
if -31.329673016395393 < n < 5667669302503732.0
1. Initial program 3.5
  \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
2. Using strategy rm
3. Applied add-cube-cbrt 3.5
  \[\leadsto \color{blue}{{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}^3}\]
Recombined 2 regimes into one program.
Removed slow pow expressions

Runtime

Please include this information when filing a bug report:

herbie --seed '#(1317229083 385665165 927243142 4133899538 3082641653 3120456537)'
(FPCore (x n)
  :name "NMSE problem 3.4.6"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))

Error

Derivation

`if n < -31.329673016395393 or 5667669302503732.0 < n`

`if -31.329673016395393 < n < 5667669302503732.0`

Runtime