\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
Test:
NMSE problem 3.4.6
Bits:
128 bits
Bits error versus x
Bits error versus n
Time: 1.6 m
Input Error: 30.7
Output Error: 8.4
Log: ⚲
\(\begin{cases} \left(\frac{\frac{1}{x}}{n} - \frac{\log x}{n \cdot \left(n \cdot x\right)}\right) - \frac{\frac{\frac{1}{2}}{n}}{{x}^2} & \text{when } n \le -2.3031253660826823 \cdot 10^{+26} \\ \sqrt[3]{{\left(\sqrt[3]{{\left(\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}^3}\right)}^3}\right)}^3} & \text{when } n \le 479155699082691.3 \\ \left(\frac{\frac{1}{x}}{n} - \frac{\log x}{n \cdot \left(n \cdot x\right)}\right) - \frac{\frac{\frac{1}{2}}{n}}{{x}^2} & \text{otherwise} \end{cases}\)

    if n < -2.3031253660826823e+26 or 479155699082691.3 < n

    1. Started with
      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
      43.6
    2. Applied taylor to get
      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leadsto \frac{1}{x \cdot n} - \left(\frac{\log x}{x \cdot {n}^2} + \frac{1}{2} \cdot \frac{1}{{x}^2 \cdot n}\right)\]
      9.4
    3. Taylor expanded around inf to get
      \[\color{red}{\frac{1}{x \cdot n} - \left(\frac{\log x}{x \cdot {n}^2} + \frac{1}{2} \cdot \frac{1}{{x}^2 \cdot n}\right)} \leadsto \color{blue}{\frac{1}{x \cdot n} - \left(\frac{\log x}{x \cdot {n}^2} + \frac{1}{2} \cdot \frac{1}{{x}^2 \cdot n}\right)}\]
      9.4
    4. Applied simplify to get
      \[\color{red}{\frac{1}{x \cdot n} - \left(\frac{\log x}{x \cdot {n}^2} + \frac{1}{2} \cdot \frac{1}{{x}^2 \cdot n}\right)} \leadsto \color{blue}{\left(\frac{\frac{1}{x}}{n} - \frac{\log x}{n \cdot \left(n \cdot x\right)}\right) - \frac{\frac{\frac{1}{2}}{n}}{{x}^2}}\]
      8.8

    if -2.3031253660826823e+26 < n < 479155699082691.3

    1. Started with
      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
      7.6
    2. Using strategy rm
      7.6
    3. Applied add-cbrt-cube to get
      \[\color{red}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \leadsto \color{blue}{\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}^3}}\]
      7.6
    4. Using strategy rm
      7.6
    5. Applied add-cbrt-cube to get
      \[\sqrt[3]{{\color{red}{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}}^3} \leadsto \sqrt[3]{{\color{blue}{\left(\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}^3}\right)}}^3}\]
      7.6
    6. Using strategy rm
      7.6
    7. Applied add-cbrt-cube to get
      \[\sqrt[3]{{\left(\sqrt[3]{{\color{red}{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}}^3}\right)}^3} \leadsto \sqrt[3]{{\left(\sqrt[3]{{\color{blue}{\left(\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}^3}\right)}}^3}\right)}^3}\]
      7.6
  1. Removed slow pow expressions

Original test:


(lambda ((x default) (n default))
  #:name "NMSE problem 3.4.6"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))