\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
Test:
NMSE problem 3.4.6
Bits:
128 bits
Bits error versus x
Bits error versus n
Time: 40.8 s
Input Error: 14.7
Output Error: 5.2
Log: ⚲
Profile: 🕒
\(\begin{cases} \left(\frac{\frac{1}{n}}{x} - \frac{\frac{1}{2}}{{x}^2 \cdot n}\right) + \frac{\log x}{x \cdot \left(n \cdot n\right)} & \text{when } n \le -46350.824f0 \\ {\left({\left(\sqrt[3]{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)}^{3}\right)}^3 & \text{when } n \le 1.00002963f+09 \\ \frac{\left(\frac{\frac{2}{x}}{n} - \frac{\frac{1}{n}}{x \cdot x}\right) - \frac{4}{n \cdot n} \cdot \frac{\log x}{x}}{{x}^{\left(\frac{1}{n}\right)} + {\left(1 + x\right)}^{\left(\frac{1}{n}\right)}} & \text{otherwise} \end{cases}\)

    if n < -46350.824f0

    1. Started with
      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
      21.1
    2. Using strategy rm
      21.1
    3. Applied add-cube-cbrt to get
      \[\color{red}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \leadsto \color{blue}{{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}^3}\]
      21.1
    4. Applied taylor to get
      \[{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}^3 \leadsto {\left(\sqrt[3]{\frac{1}{n \cdot x} - \left(\frac{\log x}{{n}^2 \cdot x} + \frac{1}{2} \cdot \frac{1}{n \cdot {x}^2}\right)}\right)}^3\]
      5.5
    5. Taylor expanded around inf to get
      \[{\left(\sqrt[3]{\color{red}{\frac{1}{n \cdot x} - \left(\frac{\log x}{{n}^2 \cdot x} + \frac{1}{2} \cdot \frac{1}{n \cdot {x}^2}\right)}}\right)}^3 \leadsto {\left(\sqrt[3]{\color{blue}{\frac{1}{n \cdot x} - \left(\frac{\log x}{{n}^2 \cdot x} + \frac{1}{2} \cdot \frac{1}{n \cdot {x}^2}\right)}}\right)}^3\]
      5.5
    6. Applied simplify to get
      \[\color{red}{{\left(\sqrt[3]{\frac{1}{n \cdot x} - \left(\frac{\log x}{{n}^2 \cdot x} + \frac{1}{2} \cdot \frac{1}{n \cdot {x}^2}\right)}\right)}^3} \leadsto \color{blue}{\left(\frac{1}{n \cdot x} - \frac{\frac{\frac{1}{2}}{n}}{x \cdot x}\right) - \frac{\frac{\log x}{n \cdot x}}{n}}\]
      5.3
    7. Applied taylor to get
      \[\left(\frac{1}{n \cdot x} - \frac{\frac{\frac{1}{2}}{n}}{x \cdot x}\right) - \frac{\frac{\log x}{n \cdot x}}{n} \leadsto \left(\frac{\log x}{{n}^2 \cdot x} + \frac{1}{n \cdot x}\right) - \frac{1}{2} \cdot \frac{1}{n \cdot {x}^2}\]
      5.1
    8. Taylor expanded around inf to get
      \[\color{red}{\left(\frac{\log x}{{n}^2 \cdot x} + \frac{1}{n \cdot x}\right) - \frac{1}{2} \cdot \frac{1}{n \cdot {x}^2}} \leadsto \color{blue}{\left(\frac{\log x}{{n}^2 \cdot x} + \frac{1}{n \cdot x}\right) - \frac{1}{2} \cdot \frac{1}{n \cdot {x}^2}}\]
      5.1
    9. Applied simplify to get
      \[\left(\frac{\log x}{{n}^2 \cdot x} + \frac{1}{n \cdot x}\right) - \frac{1}{2} \cdot \frac{1}{n \cdot {x}^2} \leadsto \left(\frac{\frac{1}{n}}{x} - \frac{\frac{1}{2}}{{x}^2 \cdot n}\right) + \frac{\log x}{x \cdot \left(n \cdot n\right)}\]
      4.1
    10. Applied final simplification

    if -46350.824f0 < n < 1.00002963f+09

    1. Started with
      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
      6.6
    2. Using strategy rm
      6.6
    3. Applied add-cube-cbrt to get
      \[\color{red}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \leadsto \color{blue}{{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}^3}\]
      6.7
    4. Using strategy rm
      6.7
    5. Applied add-cube-cbrt to get
      \[{\color{red}{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}}^3 \leadsto {\color{blue}{\left({\left(\sqrt[3]{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)}^3\right)}}^3\]
      6.7
    6. Using strategy rm
      6.7
    7. Applied pow3 to get
      \[{\color{red}{\left({\left(\sqrt[3]{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)}^3\right)}}^3 \leadsto {\color{blue}{\left({\left(\sqrt[3]{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)}^{3}\right)}}^3\]
      6.7

    if 1.00002963f+09 < n

    1. Started with
      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
      19.9
    2. Using strategy rm
      19.9
    3. Applied add-cube-cbrt to get
      \[\color{red}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \leadsto \color{blue}{{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}^3}\]
      19.9
    4. Using strategy rm
      19.9
    5. Applied add-cube-cbrt to get
      \[{\color{red}{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}}^3 \leadsto {\color{blue}{\left({\left(\sqrt[3]{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)}^3\right)}}^3\]
      19.9
    6. Using strategy rm
      19.9
    7. Applied flip-- to get
      \[{\left({\left(\sqrt[3]{\sqrt[3]{\color{red}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}}\right)}^3\right)}^3 \leadsto {\left({\left(\sqrt[3]{\sqrt[3]{\color{blue}{\frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^2 - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^2}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}}}}\right)}^3\right)}^3\]
      19.9
    8. Applied cbrt-div to get
      \[{\left({\left(\sqrt[3]{\color{red}{\sqrt[3]{\frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^2 - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^2}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}}}}\right)}^3\right)}^3 \leadsto {\left({\left(\sqrt[3]{\color{blue}{\frac{\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^2 - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^2}}{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}}}}\right)}^3\right)}^3\]
      19.9
    9. Applied cbrt-div to get
      \[{\left({\color{red}{\left(\sqrt[3]{\frac{\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^2 - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^2}}{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}}}\right)}}^3\right)}^3 \leadsto {\left({\color{blue}{\left(\frac{\sqrt[3]{\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^2 - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^2}}}{\sqrt[3]{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}}}\right)}}^3\right)}^3\]
      19.9
    10. Applied cube-div to get
      \[{\color{red}{\left({\left(\frac{\sqrt[3]{\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^2 - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^2}}}{\sqrt[3]{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}}}\right)}^3\right)}}^3 \leadsto {\color{blue}{\left(\frac{{\left(\sqrt[3]{\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^2 - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^2}}\right)}^3}{{\left(\sqrt[3]{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}}\right)}^3}\right)}}^3\]
      19.9
    11. Applied simplify to get
      \[{\left(\frac{{\left(\sqrt[3]{\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^2 - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^2}}\right)}^3}{\color{red}{{\left(\sqrt[3]{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}}\right)}^3}}\right)}^3 \leadsto {\left(\frac{{\left(\sqrt[3]{\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^2 - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^2}}\right)}^3}{\color{blue}{\sqrt[3]{{x}^{\left(\frac{1}{n}\right)} + {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}}}\right)}^3\]
      19.9
    12. Applied taylor to get
      \[{\left(\frac{{\left(\sqrt[3]{\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^2 - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^2}}\right)}^3}{\sqrt[3]{{x}^{\left(\frac{1}{n}\right)} + {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}}\right)}^3 \leadsto {\left(\frac{{\left(\sqrt[3]{\sqrt[3]{2 \cdot \frac{1}{n \cdot x} - \left(4 \cdot \frac{\log x}{{n}^2 \cdot x} + \frac{1}{n \cdot {x}^2}\right)}}\right)}^3}{\sqrt[3]{{x}^{\left(\frac{1}{n}\right)} + {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}}\right)}^3\]
      5.6
    13. Taylor expanded around inf to get
      \[{\left(\frac{{\left(\sqrt[3]{\sqrt[3]{\color{red}{2 \cdot \frac{1}{n \cdot x} - \left(4 \cdot \frac{\log x}{{n}^2 \cdot x} + \frac{1}{n \cdot {x}^2}\right)}}}\right)}^3}{\sqrt[3]{{x}^{\left(\frac{1}{n}\right)} + {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}}\right)}^3 \leadsto {\left(\frac{{\left(\sqrt[3]{\sqrt[3]{\color{blue}{2 \cdot \frac{1}{n \cdot x} - \left(4 \cdot \frac{\log x}{{n}^2 \cdot x} + \frac{1}{n \cdot {x}^2}\right)}}}\right)}^3}{\sqrt[3]{{x}^{\left(\frac{1}{n}\right)} + {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}}\right)}^3\]
      5.6
    14. Applied simplify to get
      \[{\left(\frac{{\left(\sqrt[3]{\sqrt[3]{2 \cdot \frac{1}{n \cdot x} - \left(4 \cdot \frac{\log x}{{n}^2 \cdot x} + \frac{1}{n \cdot {x}^2}\right)}}\right)}^3}{\sqrt[3]{{x}^{\left(\frac{1}{n}\right)} + {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}}\right)}^3 \leadsto \frac{\left(\frac{\frac{2}{x}}{n} - \frac{\frac{1}{n}}{x \cdot x}\right) - \frac{4}{n \cdot n} \cdot \frac{\log x}{x}}{{x}^{\left(\frac{1}{n}\right)} + {\left(1 + x\right)}^{\left(\frac{1}{n}\right)}}\]
      4.0
    15. Applied final simplification
  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))))