\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
Test:
NMSE problem 3.4.6
Bits:
128 bits
Bits error versus x
Bits error versus n
Time: 1.0 m
Input Error: 16.3
Output Error: 13.0
Log: ⚲
Profile: 🕒
\(\begin{cases} {\left(\log \left(e^{{\left(\sqrt[3]{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)}^3}\right)\right)}^3 & \text{when } n \le 66058348.0f0 \\ \left(\frac{\frac{1}{n}}{x} - \frac{\frac{1}{2}}{{x}^2 \cdot n}\right) + \frac{\log x}{x \cdot \left(n \cdot n\right)} & \text{when } n \le 2.4911614f+20 \\ \left(\left(\frac{1}{n \cdot x} + 1\right) - e^{\frac{\log x}{n}}\right) - \frac{\log x}{n} & \text{otherwise} \end{cases}\)

    if n < 66058348.0f0

    1. Started with
      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
      12.4
    2. Using strategy rm
      12.4
    3. Applied add-cube-cbrt to get
      \[\color{red}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \leadsto \color{blue}{{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}^3}\]
      12.4
    4. Using strategy rm
      12.4
    5. Applied add-log-exp to get
      \[{\color{red}{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}}^3 \leadsto {\color{blue}{\left(\log \left(e^{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)\right)}}^3\]
      12.6
    6. Using strategy rm
      12.6
    7. Applied add-cube-cbrt to get
      \[{\left(\log \left(e^{\color{red}{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}}\right)\right)}^3 \leadsto {\left(\log \left(e^{\color{blue}{{\left(\sqrt[3]{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)}^3}}\right)\right)}^3\]
      12.6

    if 66058348.0f0 < n < 2.4911614f+20

    1. Started with
      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
      23.8
    2. Using strategy rm
      23.8
    3. Applied add-cube-cbrt to get
      \[\color{red}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \leadsto \color{blue}{{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}^3}\]
      23.8
    4. Using strategy rm
      23.8
    5. Applied add-log-exp to get
      \[{\color{red}{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}}^3 \leadsto {\color{blue}{\left(\log \left(e^{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)\right)}}^3\]
      23.8
    6. Applied taylor to get
      \[{\left(\log \left(e^{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)\right)}^3 \leadsto {\left(\log \left(e^{\sqrt[3]{\frac{1}{n \cdot x} - \left(\frac{\log x}{{n}^2 \cdot x} + \frac{1}{2} \cdot \frac{1}{n \cdot {x}^2}\right)}}\right)\right)}^3\]
      18.0
    7. Taylor expanded around inf to get
      \[{\left(\log \left(e^{\sqrt[3]{\color{red}{\frac{1}{n \cdot x} - \left(\frac{\log x}{{n}^2 \cdot x} + \frac{1}{2} \cdot \frac{1}{n \cdot {x}^2}\right)}}}\right)\right)}^3 \leadsto {\left(\log \left(e^{\sqrt[3]{\color{blue}{\frac{1}{n \cdot x} - \left(\frac{\log x}{{n}^2 \cdot x} + \frac{1}{2} \cdot \frac{1}{n \cdot {x}^2}\right)}}}\right)\right)}^3\]
      18.0
    8. Applied simplify to get
      \[\color{red}{{\left(\log \left(e^{\sqrt[3]{\frac{1}{n \cdot x} - \left(\frac{\log x}{{n}^2 \cdot x} + \frac{1}{2} \cdot \frac{1}{n \cdot {x}^2}\right)}}\right)\right)}^3} \leadsto \color{blue}{\left(\frac{\frac{1}{n}}{x} - \frac{\frac{\frac{1}{2}}{n}}{x \cdot x}\right) - \frac{\frac{\log x}{n \cdot x}}{n}}\]
      2.4
    9. Applied taylor to get
      \[\left(\frac{\frac{1}{n}}{x} - \frac{\frac{\frac{1}{2}}{n}}{x \cdot x}\right) - \frac{\frac{\log x}{n \cdot x}}{n} \leadsto \left(\frac{\log x}{{n}^2 \cdot x} + \frac{1}{n \cdot x}\right) - \frac{1}{2} \cdot \frac{1}{n \cdot {x}^2}\]
      3.6
    10. Taylor expanded around inf to get
      \[\color{red}{\left(\frac{\log x}{{n}^2 \cdot x} + \frac{1}{n \cdot x}\right) - \frac{1}{2} \cdot \frac{1}{n \cdot {x}^2}} \leadsto \color{blue}{\left(\frac{\log x}{{n}^2 \cdot x} + \frac{1}{n \cdot x}\right) - \frac{1}{2} \cdot \frac{1}{n \cdot {x}^2}}\]
      3.6
    11. Applied simplify to get
      \[\left(\frac{\log x}{{n}^2 \cdot x} + \frac{1}{n \cdot x}\right) - \frac{1}{2} \cdot \frac{1}{n \cdot {x}^2} \leadsto \left(\frac{\frac{1}{n}}{x} - \frac{\frac{1}{2}}{{x}^2 \cdot n}\right) + \frac{\log x}{x \cdot \left(n \cdot n\right)}\]
      2.4
    12. Applied final simplification

    if 2.4911614f+20 < n

    1. Started with
      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
      28.7
    2. Applied taylor to get
      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leadsto \left(\left(1 + \frac{1}{n \cdot x}\right) - \frac{\log x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\]
      23.3
    3. Taylor expanded around inf to get
      \[\color{red}{\left(\left(1 + \frac{1}{n \cdot x}\right) - \frac{\log x}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leadsto \color{blue}{\left(\left(1 + \frac{1}{n \cdot x}\right) - \frac{\log x}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
      23.3
    4. Applied taylor to get
      \[\left(\left(1 + \frac{1}{n \cdot x}\right) - \frac{\log x}{n}\right) - {x}^{\left(\frac{1}{n}\right)} \leadsto \left(\left(1 + \frac{1}{n \cdot x}\right) - \frac{\log x}{n}\right) - e^{\frac{\log x}{n}}\]
      23.3
    5. Taylor expanded around 0 to get
      \[\left(\left(1 + \frac{1}{n \cdot x}\right) - \frac{\log x}{n}\right) - \color{red}{e^{\frac{\log x}{n}}} \leadsto \left(\left(1 + \frac{1}{n \cdot x}\right) - \frac{\log x}{n}\right) - \color{blue}{e^{\frac{\log x}{n}}}\]
      23.3
    6. Applied simplify to get
      \[\left(\left(1 + \frac{1}{n \cdot x}\right) - \frac{\log x}{n}\right) - e^{\frac{\log x}{n}} \leadsto \left(\left(\frac{1}{n \cdot x} + 1\right) - e^{\frac{\log x}{n}}\right) - \frac{\log x}{n}\]
      20.6
    7. 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))))