\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
Test:
NMSE problem 3.4.6
Bits:
128 bits
Bits error versus x
Bits error versus n
Time: 32.8 s
Input Error: 16.2
Output Error: 8.3
Log: ⚲
Profile: 🕒
\(\begin{cases} \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(1 - \sqrt{e^{\frac{\log x}{n}}}\right) & \text{when } x \le 2.24202f-14 \\ \left(\left(\frac{1}{n \cdot x} + 1\right) - e^{\frac{\log x}{n}}\right) - \frac{\log x}{n} & \text{when } x \le 0.9963511f0 \\ \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\left(\frac{\frac{\frac{1}{2}}{n}}{x} - \frac{\frac{1}{4}}{x \cdot n} \cdot \frac{\log x}{n}\right) - \frac{\frac{1}{4}}{{x}^2 \cdot n}\right) & \text{otherwise} \end{cases}\)

    if x < 2.24202f-14

    1. Started with
      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
      18.4
    2. Using strategy rm
      18.4
    3. Applied add-cube-cbrt to get
      \[\color{red}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \leadsto \color{blue}{{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}^3}\]
      18.4
    4. Using strategy rm
      18.4
    5. Applied add-cube-cbrt to get
      \[{\color{red}{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}}^3 \leadsto {\color{blue}{\left({\left(\sqrt[3]{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)}^3\right)}}^3\]
      18.5
    6. Using strategy rm
      18.5
    7. Applied add-sqr-sqrt to get
      \[{\left({\left(\sqrt[3]{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \color{red}{{x}^{\left(\frac{1}{n}\right)}}}}\right)}^3\right)}^3 \leadsto {\left({\left(\sqrt[3]{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \color{blue}{{\left(\sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)}^2}}}\right)}^3\right)}^3\]
      18.6
    8. Applied add-sqr-sqrt to get
      \[{\left({\left(\sqrt[3]{\sqrt[3]{\color{red}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - {\left(\sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)}^2}}\right)}^3\right)}^3 \leadsto {\left({\left(\sqrt[3]{\sqrt[3]{\color{blue}{{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right)}^2} - {\left(\sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)}^2}}\right)}^3\right)}^3\]
      18.6
    9. Applied difference-of-squares to get
      \[{\left({\left(\sqrt[3]{\sqrt[3]{\color{red}{{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right)}^2 - {\left(\sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)}^2}}}\right)}^3\right)}^3 \leadsto {\left({\left(\sqrt[3]{\sqrt[3]{\color{blue}{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)}}}\right)}^3\right)}^3\]
      18.6
    10. Applied taylor to get
      \[{\left({\left(\sqrt[3]{\sqrt[3]{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)}}\right)}^3\right)}^3 \leadsto {\left({\left(\sqrt[3]{\sqrt[3]{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(1 - \sqrt{e^{\frac{\log x}{n}}}\right)}}\right)}^3\right)}^3\]
      18.6
    11. Taylor expanded around 0 to get
      \[{\left({\left(\sqrt[3]{\sqrt[3]{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \color{red}{\left(1 - \sqrt{e^{\frac{\log x}{n}}}\right)}}}\right)}^3\right)}^3 \leadsto {\left({\left(\sqrt[3]{\sqrt[3]{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \color{blue}{\left(1 - \sqrt{e^{\frac{\log x}{n}}}\right)}}}\right)}^3\right)}^3\]
      18.6
    12. Applied simplify to get
      \[{\left({\left(\sqrt[3]{\sqrt[3]{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(1 - \sqrt{e^{\frac{\log x}{n}}}\right)}}\right)}^3\right)}^3 \leadsto \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(1 - \sqrt{e^{\frac{\log x}{n}}}\right)\]
      18.6
    13. Applied final simplification

    if 2.24202f-14 < x < 0.9963511f0

    1. Started with
      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
      28.5
    2. Applied taylor to get
      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leadsto \left(\left(1 + \frac{1}{n \cdot x}\right) - \frac{\log x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\]
      14.3
    3. Taylor expanded around inf to get
      \[\color{red}{\left(\left(1 + \frac{1}{n \cdot x}\right) - \frac{\log x}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leadsto \color{blue}{\left(\left(1 + \frac{1}{n \cdot x}\right) - \frac{\log x}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
      14.3
    4. Applied taylor to get
      \[\left(\left(1 + \frac{1}{n \cdot x}\right) - \frac{\log x}{n}\right) - {x}^{\left(\frac{1}{n}\right)} \leadsto \left(\left(1 + \frac{1}{n \cdot x}\right) - \frac{\log x}{n}\right) - e^{\frac{\log x}{n}}\]
      14.3
    5. Taylor expanded around 0 to get
      \[\left(\left(1 + \frac{1}{n \cdot x}\right) - \frac{\log x}{n}\right) - \color{red}{e^{\frac{\log x}{n}}} \leadsto \left(\left(1 + \frac{1}{n \cdot x}\right) - \frac{\log x}{n}\right) - \color{blue}{e^{\frac{\log x}{n}}}\]
      14.3
    6. Applied simplify to get
      \[\left(\left(1 + \frac{1}{n \cdot x}\right) - \frac{\log x}{n}\right) - e^{\frac{\log x}{n}} \leadsto \left(\left(\frac{1}{n \cdot x} + 1\right) - e^{\frac{\log x}{n}}\right) - \frac{\log x}{n}\]
      13.6
    7. Applied final simplification

    if 0.9963511f0 < x

    1. Started with
      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
      10.7
    2. Using strategy rm
      10.7
    3. Applied add-cube-cbrt to get
      \[\color{red}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \leadsto \color{blue}{{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}^3}\]
      10.7
    4. Using strategy rm
      10.7
    5. Applied add-cube-cbrt to get
      \[{\color{red}{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}}^3 \leadsto {\color{blue}{\left({\left(\sqrt[3]{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)}^3\right)}}^3\]
      10.7
    6. Using strategy rm
      10.7
    7. Applied add-sqr-sqrt to get
      \[{\left({\left(\sqrt[3]{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \color{red}{{x}^{\left(\frac{1}{n}\right)}}}}\right)}^3\right)}^3 \leadsto {\left({\left(\sqrt[3]{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \color{blue}{{\left(\sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)}^2}}}\right)}^3\right)}^3\]
      11.3
    8. Applied add-sqr-sqrt to get
      \[{\left({\left(\sqrt[3]{\sqrt[3]{\color{red}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - {\left(\sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)}^2}}\right)}^3\right)}^3 \leadsto {\left({\left(\sqrt[3]{\sqrt[3]{\color{blue}{{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right)}^2} - {\left(\sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)}^2}}\right)}^3\right)}^3\]
      10.7
    9. Applied difference-of-squares to get
      \[{\left({\left(\sqrt[3]{\sqrt[3]{\color{red}{{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right)}^2 - {\left(\sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)}^2}}}\right)}^3\right)}^3 \leadsto {\left({\left(\sqrt[3]{\sqrt[3]{\color{blue}{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)}}}\right)}^3\right)}^3\]
      10.7
    10. Applied taylor to get
      \[{\left({\left(\sqrt[3]{\sqrt[3]{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)}}\right)}^3\right)}^3 \leadsto {\left({\left(\sqrt[3]{\sqrt[3]{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\frac{1}{2} \cdot \frac{1}{n \cdot x} - \left(\frac{1}{4} \cdot \frac{\log x}{{n}^2 \cdot x} + \frac{1}{4} \cdot \frac{1}{n \cdot {x}^2}\right)\right)}}\right)}^3\right)}^3\]
      6.0
    11. Taylor expanded around inf to get
      \[{\left({\left(\sqrt[3]{\sqrt[3]{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \color{red}{\left(\frac{1}{2} \cdot \frac{1}{n \cdot x} - \left(\frac{1}{4} \cdot \frac{\log x}{{n}^2 \cdot x} + \frac{1}{4} \cdot \frac{1}{n \cdot {x}^2}\right)\right)}}}\right)}^3\right)}^3 \leadsto {\left({\left(\sqrt[3]{\sqrt[3]{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{n \cdot x} - \left(\frac{1}{4} \cdot \frac{\log x}{{n}^2 \cdot x} + \frac{1}{4} \cdot \frac{1}{n \cdot {x}^2}\right)\right)}}}\right)}^3\right)}^3\]
      6.0
    12. Applied simplify to get
      \[{\left({\left(\sqrt[3]{\sqrt[3]{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\frac{1}{2} \cdot \frac{1}{n \cdot x} - \left(\frac{1}{4} \cdot \frac{\log x}{{n}^2 \cdot x} + \frac{1}{4} \cdot \frac{1}{n \cdot {x}^2}\right)\right)}}\right)}^3\right)}^3 \leadsto \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\left(\frac{\frac{\frac{1}{2}}{n}}{x} - \frac{\frac{1}{4}}{x \cdot n} \cdot \frac{\log x}{n}\right) - \frac{\frac{1}{4}}{{x}^2 \cdot n}\right)\]
      0.5
    13. 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))))