\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
Test:
NMSE problem 3.4.6
Bits:
128 bits
Bits error versus x
Bits error versus n
Time: 32.9 s
Input Error: 30.0
Output Error: 1.9
Log: ⚲
Profile: 🕒
\(\begin{cases} \left(\frac{1}{n \cdot x} - \frac{\frac{\log x}{n \cdot x}}{n}\right) - \frac{\frac{\frac{1}{2}}{x}}{n \cdot x} & \text{when } n \le -91310770552578.69 \\ \sqrt[3]{{\left(\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}^3}\right)}^3} & \text{when } n \le 60006574336.140175 \\ \left(\frac{1}{n \cdot x} - \frac{\frac{\log x}{n \cdot x}}{n}\right) - \frac{\frac{\frac{1}{2}}{x}}{n \cdot x} & \text{otherwise} \end{cases}\)

    if n < -91310770552578.69 or 60006574336.140175 < n

    1. Started with
      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
      43.3
    2. Using strategy rm
      43.3
    3. Applied add-cbrt-cube to get
      \[\color{red}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \leadsto \color{blue}{\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}^3}}\]
      43.3
    4. Applied taylor to get
      \[\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}^3} \leadsto \sqrt[3]{{\left(\frac{1}{n \cdot x} - \left(\frac{\log x}{{n}^2 \cdot x} + \frac{1}{2} \cdot \frac{1}{n \cdot {x}^2}\right)\right)}^3}\]
      31.3
    5. Taylor expanded around inf to get
      \[\sqrt[3]{{\color{red}{\left(\frac{1}{n \cdot x} - \left(\frac{\log x}{{n}^2 \cdot x} + \frac{1}{2} \cdot \frac{1}{n \cdot {x}^2}\right)\right)}}^3} \leadsto \sqrt[3]{{\color{blue}{\left(\frac{1}{n \cdot x} - \left(\frac{\log x}{{n}^2 \cdot x} + \frac{1}{2} \cdot \frac{1}{n \cdot {x}^2}\right)\right)}}^3}\]
      31.3
    6. Applied simplify to get
      \[\color{red}{\sqrt[3]{{\left(\frac{1}{n \cdot x} - \left(\frac{\log x}{{n}^2 \cdot x} + \frac{1}{2} \cdot \frac{1}{n \cdot {x}^2}\right)\right)}^3}} \leadsto \color{blue}{\left(\frac{1}{n \cdot x} - \frac{\frac{\log x}{n \cdot x}}{n}\right) - \frac{\frac{\frac{1}{2}}{x}}{n \cdot x}}\]
      0.8

    if -91310770552578.69 < n < 60006574336.140175

    1. Started with
      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
      3.9
    2. Using strategy rm
      3.9
    3. Applied add-cbrt-cube to get
      \[\color{red}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \leadsto \color{blue}{\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}^3}}\]
      4.0
    4. Using strategy rm
      4.0
    5. Applied add-cbrt-cube to get
      \[\sqrt[3]{{\color{red}{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}}^3} \leadsto \sqrt[3]{{\color{blue}{\left(\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}^3}\right)}}^3}\]
      4.0
  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))))