\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
Test:
NMSE problem 3.4.6
Bits:
128 bits
Bits error versus x
Bits error versus n
Time: 20.1 s
Input Error: 14.4
Output Error: 5.4
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{1}{n \cdot x} - \left(\frac{\log x}{{n}^2 \cdot x} + \frac{1}{2} \cdot \frac{1}{n \cdot {x}^2}\right) & \text{when } n \le -36550324.0f0 \\ {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - e^{\frac{\log x}{n}} & \text{when } n \le 150622.2f0 \\ \frac{1}{n \cdot x} - \left(\frac{\log x}{{n}^2 \cdot x} + \frac{1}{2} \cdot \frac{1}{n \cdot {x}^2}\right) & \text{otherwise} \end{cases}\)

    if n < -36550324.0f0 or 150622.2f0 < n

    1. Started with
      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
      20.6
    2. Applied taylor to get
      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leadsto \frac{1}{n \cdot x} - \left(\frac{\log x}{{n}^2 \cdot x} + \frac{1}{2} \cdot \frac{1}{n \cdot {x}^2}\right)\]
      4.9
    3. Taylor expanded around inf to get
      \[\color{red}{\frac{1}{n \cdot x} - \left(\frac{\log x}{{n}^2 \cdot x} + \frac{1}{2} \cdot \frac{1}{n \cdot {x}^2}\right)} \leadsto \color{blue}{\frac{1}{n \cdot x} - \left(\frac{\log x}{{n}^2 \cdot x} + \frac{1}{2} \cdot \frac{1}{n \cdot {x}^2}\right)}\]
      4.9

    if -36550324.0f0 < n < 150622.2f0

    1. Started with
      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
      5.8
    2. Applied taylor to get
      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leadsto {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - e^{\frac{\log x}{n}}\]
      6.0
    3. Taylor expanded around 0 to get
      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \color{red}{e^{\frac{\log x}{n}}} \leadsto {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \color{blue}{e^{\frac{\log x}{n}}}\]
      6.0
  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))))