\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
Test:
NMSE problem 3.4.6
Bits:
128 bits
Bits error versus x
Bits error versus n
Time: 11.6 s
Input Error: 28.3
Output Error: 16.2
Log: ⚲
Profile: 🕒
\((\left(\frac{\log_* (1 + \frac{1}{x})}{n}\right) * \left((\left(\frac{\log_* (1 + \frac{1}{x})}{n}\right) * \frac{1}{2} + 1)_*\right) + \left(1 - {x}^{\left(\frac{1}{n}\right)}\right))_*\)
  1. Started with
    \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
    28.3
  2. Using strategy rm
    28.3
  3. Applied add-exp-log to get
    \[{\color{red}{\left(x + 1\right)}}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leadsto {\color{blue}{\left(e^{\log \left(x + 1\right)}\right)}}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
    28.3
  4. Applied pow-exp to get
    \[\color{red}{{\left(e^{\log \left(x + 1\right)}\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{1}{n}\right)} \leadsto \color{blue}{e^{\log \left(x + 1\right) \cdot \frac{1}{n}}} - {x}^{\left(\frac{1}{n}\right)}\]
    28.3
  5. Applied simplify to get
    \[e^{\color{red}{\log \left(x + 1\right) \cdot \frac{1}{n}}} - {x}^{\left(\frac{1}{n}\right)} \leadsto e^{\color{blue}{\frac{\log_* (1 + x)}{n}}} - {x}^{\left(\frac{1}{n}\right)}\]
    28.3
  6. Applied taylor to get
    \[e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)} \leadsto \left(\frac{\log_* (1 + \frac{1}{x})}{n} + \left(1 + \frac{1}{2} \cdot \frac{{\left(\log_* (1 + \frac{1}{x})\right)}^2}{{n}^2}\right)\right) - {x}^{\left(\frac{1}{n}\right)}\]
    18.8
  7. Taylor expanded around inf to get
    \[\color{red}{\left(\frac{\log_* (1 + \frac{1}{x})}{n} + \left(1 + \frac{1}{2} \cdot \frac{{\left(\log_* (1 + \frac{1}{x})\right)}^2}{{n}^2}\right)\right)} - {x}^{\left(\frac{1}{n}\right)} \leadsto \color{blue}{\left(\frac{\log_* (1 + \frac{1}{x})}{n} + \left(1 + \frac{1}{2} \cdot \frac{{\left(\log_* (1 + \frac{1}{x})\right)}^2}{{n}^2}\right)\right)} - {x}^{\left(\frac{1}{n}\right)}\]
    18.8
  8. Applied simplify to get
    \[\color{red}{\left(\frac{\log_* (1 + \frac{1}{x})}{n} + \left(1 + \frac{1}{2} \cdot \frac{{\left(\log_* (1 + \frac{1}{x})\right)}^2}{{n}^2}\right)\right) - {x}^{\left(\frac{1}{n}\right)}} \leadsto \color{blue}{(\left(\frac{\log_* (1 + \frac{1}{x})}{n}\right) * \left((\left(\frac{\log_* (1 + \frac{1}{x})}{n}\right) * \frac{1}{2} + 1)_*\right) + \left(1 - {x}^{\left(\frac{1}{n}\right)}\right))_*}\]
    16.2

Original test:


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