\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
Test:
NMSE problem 3.4.6
Bits:
128 bits
Bits error versus x
Bits error versus n
Time: 40.8 s
Input Error: 14.9
Output Error: 4.7
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{\log x}{x \cdot \left(n \cdot n\right)} + \left(\frac{1}{n \cdot x} - \frac{\frac{\frac{1}{2}}{n}}{{x}^2}\right) & \text{when } n \le -18.617496f0 \\ {\left({\left(\sqrt[3]{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)}^3\right)}^3 & \text{when } n \le 2506.933f0 \\ \frac{\log x}{x \cdot \left(n \cdot n\right)} + \left(\frac{1}{n \cdot x} - \frac{\frac{\frac{1}{2}}{n}}{{x}^2}\right) & \text{when } n \le 4.0715707f+34 \\ \frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^3 - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^3}{{\left(\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{n}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot {x}^{\left(\frac{1}{n}\right)}\right)}\right)}^3} & \text{otherwise} \end{cases}\)

    if n < -18.617496f0 or 2506.933f0 < n < 4.0715707f+34

    1. Started with
      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
      21.3
    2. Applied taylor to get
      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leadsto \frac{1}{n \cdot x} - \left(\frac{\log x}{{n}^2 \cdot x} + \frac{1}{2} \cdot \frac{1}{n \cdot {x}^2}\right)\]
      5.4
    3. Taylor expanded around inf to get
      \[\color{red}{\frac{1}{n \cdot x} - \left(\frac{\log x}{{n}^2 \cdot x} + \frac{1}{2} \cdot \frac{1}{n \cdot {x}^2}\right)} \leadsto \color{blue}{\frac{1}{n \cdot x} - \left(\frac{\log x}{{n}^2 \cdot x} + \frac{1}{2} \cdot \frac{1}{n \cdot {x}^2}\right)}\]
      5.4
    4. Applied taylor to get
      \[\frac{1}{n \cdot x} - \left(\frac{\log x}{{n}^2 \cdot x} + \frac{1}{2} \cdot \frac{1}{n \cdot {x}^2}\right) \leadsto \frac{1}{n \cdot x} - \left(\frac{1}{2} \cdot \frac{1}{n \cdot {x}^2} - \frac{\log x}{{n}^2 \cdot x}\right)\]
      4.5
    5. Taylor expanded around inf to get
      \[\frac{1}{n \cdot x} - \color{red}{\left(\frac{1}{2} \cdot \frac{1}{n \cdot {x}^2} - \frac{\log x}{{n}^2 \cdot x}\right)} \leadsto \frac{1}{n \cdot x} - \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{n \cdot {x}^2} - \frac{\log x}{{n}^2 \cdot x}\right)}\]
      4.5
    6. Applied simplify to get
      \[\frac{1}{n \cdot x} - \left(\frac{1}{2} \cdot \frac{1}{n \cdot {x}^2} - \frac{\log x}{{n}^2 \cdot x}\right) \leadsto \left(\frac{\log x}{\left(n \cdot x\right) \cdot n} + \frac{\frac{1}{n}}{x}\right) - \frac{\frac{1}{2}}{\left(x \cdot x\right) \cdot n}\]
      3.7
    7. Applied final simplification
    8. Applied simplify to get
      \[\color{red}{\left(\frac{\log x}{\left(n \cdot x\right) \cdot n} + \frac{\frac{1}{n}}{x}\right) - \frac{\frac{1}{2}}{\left(x \cdot x\right) \cdot n}} \leadsto \color{blue}{\frac{\log x}{x \cdot \left(n \cdot n\right)} + \left(\frac{1}{n \cdot x} - \frac{\frac{\frac{1}{2}}{n}}{{x}^2}\right)}\]
      4.5

    if -18.617496f0 < n < 2506.933f0

    1. Started with
      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
      3.8
    2. Using strategy rm
      3.8
    3. Applied add-cube-cbrt to get
      \[\color{red}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \leadsto \color{blue}{{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}^3}\]
      3.8
    4. Using strategy rm
      3.8
    5. Applied add-cube-cbrt to get
      \[{\color{red}{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}}^3 \leadsto {\color{blue}{\left({\left(\sqrt[3]{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)}^3\right)}}^3\]
      3.8

    if 4.0715707f+34 < n

    1. Started with
      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
      16.7
    2. Using strategy rm
      16.7
    3. Applied add-cube-cbrt to get
      \[\color{red}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \leadsto \color{blue}{{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}^3}\]
      16.7
    4. Using strategy rm
      16.7
    5. Applied flip3-- to get
      \[{\left(\sqrt[3]{\color{red}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)}^3 \leadsto {\left(\sqrt[3]{\color{blue}{\frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^{3} - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^{3}}{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{n}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot {x}^{\left(\frac{1}{n}\right)}\right)}}}\right)}^3\]
      16.7
    6. Applied cbrt-div to get
      \[{\color{red}{\left(\sqrt[3]{\frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^{3} - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^{3}}{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{n}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot {x}^{\left(\frac{1}{n}\right)}\right)}}\right)}}^3 \leadsto {\color{blue}{\left(\frac{\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^{3} - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^{3}}}{\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{n}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot {x}^{\left(\frac{1}{n}\right)}\right)}}\right)}}^3\]
      16.7
    7. Applied cube-div to get
      \[\color{red}{{\left(\frac{\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^{3} - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^{3}}}{\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{n}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot {x}^{\left(\frac{1}{n}\right)}\right)}}\right)}^3} \leadsto \color{blue}{\frac{{\left(\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^{3} - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^{3}}\right)}^3}{{\left(\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{n}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot {x}^{\left(\frac{1}{n}\right)}\right)}\right)}^3}}\]
      16.7
    8. Applied simplify to get
      \[\frac{\color{red}{{\left(\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^{3} - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^{3}}\right)}^3}}{{\left(\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{n}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot {x}^{\left(\frac{1}{n}\right)}\right)}\right)}^3} \leadsto \frac{\color{blue}{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^3 - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^3}}{{\left(\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{n}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} \cdot {x}^{\left(\frac{1}{n}\right)}\right)}\right)}^3}\]
      16.7
  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))))