\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
Test:
NMSE problem 3.4.6
Bits:
128 bits
Bits error versus x
Bits error versus n
Time: 19.3 s
Input Error: 14.8
Output Error: 14.9
Log: ⚲
Profile: 🕒
\({\left(\log \left(e^{{\left(\sqrt[3]{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)}^3}\right)\right)}^3\)
  1. Started with
    \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
    14.8
  2. Using strategy rm
    14.8
  3. Applied add-cube-cbrt to get
    \[\color{red}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \leadsto \color{blue}{{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}^3}\]
    14.8
  4. Using strategy rm
    14.8
  5. Applied add-log-exp to get
    \[{\color{red}{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}}^3 \leadsto {\color{blue}{\left(\log \left(e^{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)\right)}}^3\]
    14.9
  6. Using strategy rm
    14.9
  7. Applied add-cube-cbrt to get
    \[{\left(\log \left(e^{\color{red}{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}}\right)\right)}^3 \leadsto {\left(\log \left(e^{\color{blue}{{\left(\sqrt[3]{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)}^3}}\right)\right)}^3\]
    14.9

Original test:


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