\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
Test:
NMSE problem 3.4.6
Bits:
128 bits
Bits error versus x
Bits error versus n
Time: 39.1 s
Input Error: 15.9
Output Error: 4.6
Log: ⚲
Profile: 🕒
\(\begin{cases} \left(\left(\frac{1}{n \cdot x} + 1\right) - e^{\frac{\log x}{n}}\right) - \frac{\log x}{n} & \text{when } n \le -1.0672306f+27 \\ \left(\left(\frac{\frac{\frac{1}{2}}{x}}{n} - \frac{\frac{1}{4}}{n \cdot n} \cdot \frac{\log x}{x}\right) - \frac{\frac{1}{4}}{n \cdot {x}^2}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) & \text{when } n \le -29453316.0f0 \\ e^{\frac{x + \left(\frac{1}{3} \cdot x - \frac{1}{2}\right) \cdot {x}^2}{n}} - {x}^{\left(\frac{1}{n}\right)} & \text{when } n \le 7393.617f0 \\ \left(\left(\frac{\frac{\frac{1}{2}}{x}}{n} - \frac{\frac{1}{4}}{n \cdot n} \cdot \frac{\log x}{x}\right) - \frac{\frac{1}{4}}{n \cdot {x}^2}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) & \text{otherwise} \end{cases}\)

    if n < -1.0672306f+27

    1. Started with
      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
      28.2
    2. Applied taylor to get
      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leadsto \left(\left(1 + \frac{1}{n \cdot x}\right) - \frac{\log x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\]
      24.2
    3. Taylor expanded around inf to get
      \[\color{red}{\left(\left(1 + \frac{1}{n \cdot x}\right) - \frac{\log x}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leadsto \color{blue}{\left(\left(1 + \frac{1}{n \cdot x}\right) - \frac{\log x}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
      24.2
    4. Applied taylor to get
      \[\left(\left(1 + \frac{1}{n \cdot x}\right) - \frac{\log x}{n}\right) - {x}^{\left(\frac{1}{n}\right)} \leadsto \left(\left(1 + \frac{1}{n \cdot x}\right) - \frac{\log x}{n}\right) - e^{\frac{\log x}{n}}\]
      24.2
    5. Taylor expanded around 0 to get
      \[\left(\left(1 + \frac{1}{n \cdot x}\right) - \frac{\log x}{n}\right) - \color{red}{e^{\frac{\log x}{n}}} \leadsto \left(\left(1 + \frac{1}{n \cdot x}\right) - \frac{\log x}{n}\right) - \color{blue}{e^{\frac{\log x}{n}}}\]
      24.2
    6. Applied simplify to get
      \[\left(\left(1 + \frac{1}{n \cdot x}\right) - \frac{\log x}{n}\right) - e^{\frac{\log x}{n}} \leadsto \left(\left(\frac{1}{n \cdot x} + 1\right) - e^{\frac{\log x}{n}}\right) - \frac{\log x}{n}\]
      21.7
    7. Applied final simplification

    if -1.0672306f+27 < n < -29453316.0f0 or 7393.617f0 < n

    1. Started with
      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
      21.8
    2. Using strategy rm
      21.8
    3. Applied add-cube-cbrt to get
      \[\color{red}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \leadsto \color{blue}{{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}^3}\]
      21.8
    4. Using strategy rm
      21.8
    5. Applied add-sqr-sqrt to get
      \[{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \color{red}{{x}^{\left(\frac{1}{n}\right)}}}\right)}^3 \leadsto {\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \color{blue}{{\left(\sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)}^2}}\right)}^3\]
      22.1
    6. Applied add-sqr-sqrt to get
      \[{\left(\sqrt[3]{\color{red}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - {\left(\sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)}^2}\right)}^3 \leadsto {\left(\sqrt[3]{\color{blue}{{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right)}^2} - {\left(\sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)}^2}\right)}^3\]
      21.8
    7. Applied difference-of-squares to get
      \[{\left(\sqrt[3]{\color{red}{{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right)}^2 - {\left(\sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)}^2}}\right)}^3 \leadsto {\left(\sqrt[3]{\color{blue}{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)}}\right)}^3\]
      21.8
    8. Applied cbrt-prod to get
      \[{\color{red}{\left(\sqrt[3]{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)}\right)}}^3 \leadsto {\color{blue}{\left(\sqrt[3]{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}} \cdot \sqrt[3]{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}}\right)}}^3\]
      21.8
    9. Applied taylor to get
      \[{\left(\sqrt[3]{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}} \cdot \sqrt[3]{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}}\right)}^3 \leadsto {\left(\sqrt[3]{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}} \cdot \sqrt[3]{\frac{1}{2} \cdot \frac{1}{n \cdot x} - \left(\frac{1}{4} \cdot \frac{\log x}{{n}^2 \cdot x} + \frac{1}{4} \cdot \frac{1}{n \cdot {x}^2}\right)}\right)}^3\]
      4.9
    10. Taylor expanded around inf to get
      \[{\left(\sqrt[3]{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}} \cdot \sqrt[3]{\color{red}{\frac{1}{2} \cdot \frac{1}{n \cdot x} - \left(\frac{1}{4} \cdot \frac{\log x}{{n}^2 \cdot x} + \frac{1}{4} \cdot \frac{1}{n \cdot {x}^2}\right)}}\right)}^3 \leadsto {\left(\sqrt[3]{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}} \cdot \sqrt[3]{\color{blue}{\frac{1}{2} \cdot \frac{1}{n \cdot x} - \left(\frac{1}{4} \cdot \frac{\log x}{{n}^2 \cdot x} + \frac{1}{4} \cdot \frac{1}{n \cdot {x}^2}\right)}}\right)}^3\]
      4.9
    11. Applied simplify to get
      \[{\left(\sqrt[3]{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}} \cdot \sqrt[3]{\frac{1}{2} \cdot \frac{1}{n \cdot x} - \left(\frac{1}{4} \cdot \frac{\log x}{{n}^2 \cdot x} + \frac{1}{4} \cdot \frac{1}{n \cdot {x}^2}\right)}\right)}^3 \leadsto \left(\left(\frac{\frac{\frac{1}{2}}{x}}{n} - \frac{\frac{1}{4}}{n \cdot n} \cdot \frac{\log x}{x}\right) - \frac{\frac{1}{4}}{n \cdot {x}^2}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)\]
      3.7
    12. Applied final simplification

    if -29453316.0f0 < n < 7393.617f0

    1. Started with
      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
      5.0
    2. Using strategy rm
      5.0
    3. Applied add-cube-cbrt to get
      \[\color{red}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \leadsto \color{blue}{{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}^3}\]
      5.1
    4. Using strategy rm
      5.1
    5. Applied add-exp-log to get
      \[{\left(\sqrt[3]{{\color{red}{\left(x + 1\right)}}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}^3 \leadsto {\left(\sqrt[3]{{\color{blue}{\left(e^{\log \left(x + 1\right)}\right)}}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}^3\]
      5.1
    6. Applied pow-exp to get
      \[{\left(\sqrt[3]{\color{red}{{\left(e^{\log \left(x + 1\right)}\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{1}{n}\right)}}\right)}^3 \leadsto {\left(\sqrt[3]{\color{blue}{e^{\log \left(x + 1\right) \cdot \frac{1}{n}}} - {x}^{\left(\frac{1}{n}\right)}}\right)}^3\]
      5.1
    7. Applied simplify to get
      \[{\left(\sqrt[3]{e^{\color{red}{\log \left(x + 1\right) \cdot \frac{1}{n}}} - {x}^{\left(\frac{1}{n}\right)}}\right)}^3 \leadsto {\left(\sqrt[3]{e^{\color{blue}{\frac{\log \left(x + 1\right)}{n}}} - {x}^{\left(\frac{1}{n}\right)}}\right)}^3\]
      5.1
    8. Applied taylor to get
      \[{\left(\sqrt[3]{e^{\frac{\log \left(x + 1\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}}\right)}^3 \leadsto {\left(\sqrt[3]{e^{\frac{\left(\frac{1}{3} \cdot {x}^{3} + x\right) - \frac{1}{2} \cdot {x}^2}{n}} - {x}^{\left(\frac{1}{n}\right)}}\right)}^3\]
      9.8
    9. Taylor expanded around 0 to get
      \[{\left(\sqrt[3]{e^{\frac{\color{red}{\left(\frac{1}{3} \cdot {x}^{3} + x\right) - \frac{1}{2} \cdot {x}^2}}{n}} - {x}^{\left(\frac{1}{n}\right)}}\right)}^3 \leadsto {\left(\sqrt[3]{e^{\frac{\color{blue}{\left(\frac{1}{3} \cdot {x}^{3} + x\right) - \frac{1}{2} \cdot {x}^2}}{n}} - {x}^{\left(\frac{1}{n}\right)}}\right)}^3\]
      9.8
    10. Applied simplify to get
      \[{\left(\sqrt[3]{e^{\frac{\left(\frac{1}{3} \cdot {x}^{3} + x\right) - \frac{1}{2} \cdot {x}^2}{n}} - {x}^{\left(\frac{1}{n}\right)}}\right)}^3 \leadsto e^{\frac{x + \left(\frac{1}{3} \cdot {x}^3 - \frac{1}{2} \cdot \left(x \cdot x\right)\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\]
      9.7
    11. Applied final simplification
    12. Applied simplify to get
      \[\color{red}{e^{\frac{x + \left(\frac{1}{3} \cdot {x}^3 - \frac{1}{2} \cdot \left(x \cdot x\right)\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}} \leadsto \color{blue}{e^{\frac{x + \left(\frac{1}{3} \cdot x - \frac{1}{2}\right) \cdot {x}^2}{n}} - {x}^{\left(\frac{1}{n}\right)}}\]
      0.9
  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))))