\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
Test:
NMSE problem 3.4.6
Bits:
128 bits
Bits error versus x
Bits error versus n
Time: 40.2 s
Input Error: 18.6
Output Error: 10.6
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{1 - e^{\frac{\log x}{n} + \frac{\log x}{n}}}{{x}^{\left(\frac{1}{n}\right)} + {\left(1 + x\right)}^{\left(\frac{1}{n}\right)}} & \text{when } x \le 8.616085f-19 \\ \left(\left(\frac{1}{n \cdot x} + 1\right) - e^{\frac{\log x}{n}}\right) - \frac{\log x}{n} & \text{when } x \le 0.036267824f0 \\ {\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right)}^2 - {x}^{\left(\frac{1}{n}\right)} & \text{when } x \le 2302.4204f0 \\ \frac{\log x}{x \cdot \left(n \cdot n\right)} + \left(\frac{1}{n \cdot x} - \frac{\frac{\frac{1}{2}}{n}}{{x}^2}\right) & \text{when } x \le 3.8668015f+18 \\ \frac{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}\right)}{{x}^{\left(\frac{1}{n}\right)} + 1} & \text{otherwise} \end{cases}\)

    if x < 8.616085f-19

    1. Started with
      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
      19.0
    2. Using strategy rm
      19.0
    3. Applied add-cube-cbrt to get
      \[\color{red}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \leadsto \color{blue}{{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}^3}\]
      19.0
    4. Using strategy rm
      19.0
    5. Applied add-cube-cbrt to get
      \[{\color{red}{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}}^3 \leadsto {\color{blue}{\left({\left(\sqrt[3]{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)}^3\right)}}^3\]
      19.0
    6. Using strategy rm
      19.0
    7. Applied flip-- to get
      \[{\left({\left(\sqrt[3]{\sqrt[3]{\color{red}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}}\right)}^3\right)}^3 \leadsto {\left({\left(\sqrt[3]{\sqrt[3]{\color{blue}{\frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^2 - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^2}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}}}}\right)}^3\right)}^3\]
      19.0
    8. Applied cbrt-div to get
      \[{\left({\left(\sqrt[3]{\color{red}{\sqrt[3]{\frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^2 - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^2}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}}}}\right)}^3\right)}^3 \leadsto {\left({\left(\sqrt[3]{\color{blue}{\frac{\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^2 - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^2}}{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}}}}\right)}^3\right)}^3\]
      19.1
    9. Applied taylor to get
      \[{\left({\left(\sqrt[3]{\frac{\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^2 - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^2}}{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}}}\right)}^3\right)}^3 \leadsto {\left({\left(\sqrt[3]{\frac{\sqrt[3]{1 - {\left(e^{\frac{\log x}{n}}\right)}^2}}{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}}}\right)}^3\right)}^3\]
      19.1
    10. Taylor expanded around 0 to get
      \[{\left({\left(\sqrt[3]{\frac{\sqrt[3]{\color{red}{1 - {\left(e^{\frac{\log x}{n}}\right)}^2}}}{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}}}\right)}^3\right)}^3 \leadsto {\left({\left(\sqrt[3]{\frac{\sqrt[3]{\color{blue}{1 - {\left(e^{\frac{\log x}{n}}\right)}^2}}}{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}}}\right)}^3\right)}^3\]
      19.1
    11. Applied simplify to get
      \[{\left({\left(\sqrt[3]{\frac{\sqrt[3]{1 - {\left(e^{\frac{\log x}{n}}\right)}^2}}{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}}}\right)}^3\right)}^3 \leadsto \frac{1 - e^{\frac{\log x}{n} + \frac{\log x}{n}}}{{x}^{\left(\frac{1}{n}\right)} + {\left(1 + x\right)}^{\left(\frac{1}{n}\right)}}\]
      18.9
    12. Applied final simplification

    if 8.616085f-19 < x < 0.036267824f0

    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
      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leadsto \left(\left(1 + \frac{1}{n \cdot x}\right) - \frac{\log x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\]
      13.1
    3. Taylor expanded around inf to get
      \[\color{red}{\left(\left(1 + \frac{1}{n \cdot x}\right) - \frac{\log x}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leadsto \color{blue}{\left(\left(1 + \frac{1}{n \cdot x}\right) - \frac{\log x}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
      13.1
    4. Applied taylor to get
      \[\left(\left(1 + \frac{1}{n \cdot x}\right) - \frac{\log x}{n}\right) - {x}^{\left(\frac{1}{n}\right)} \leadsto \left(\left(1 + \frac{1}{n \cdot x}\right) - \frac{\log x}{n}\right) - e^{\frac{\log x}{n}}\]
      13.1
    5. Taylor expanded around 0 to get
      \[\left(\left(1 + \frac{1}{n \cdot x}\right) - \frac{\log x}{n}\right) - \color{red}{e^{\frac{\log x}{n}}} \leadsto \left(\left(1 + \frac{1}{n \cdot x}\right) - \frac{\log x}{n}\right) - \color{blue}{e^{\frac{\log x}{n}}}\]
      13.1
    6. Applied simplify to get
      \[\left(\left(1 + \frac{1}{n \cdot x}\right) - \frac{\log x}{n}\right) - e^{\frac{\log x}{n}} \leadsto \left(\left(\frac{1}{n \cdot x} + 1\right) - e^{\frac{\log x}{n}}\right) - \frac{\log x}{n}\]
      12.5
    7. Applied final simplification

    if 0.036267824f0 < x < 2302.4204f0

    1. Started with
      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
      17.9
    2. Using strategy rm
      17.9
    3. Applied add-sqr-sqrt to get
      \[\color{red}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{1}{n}\right)} \leadsto \color{blue}{{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right)}^2} - {x}^{\left(\frac{1}{n}\right)}\]
      18.0

    if 2302.4204f0 < x < 3.8668015f+18

    1. Started with
      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
      24.6
    2. Applied taylor to get
      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leadsto \frac{1}{n \cdot x} - \left(\frac{\log x}{{n}^2 \cdot x} + \frac{1}{2} \cdot \frac{1}{n \cdot {x}^2}\right)\]
      16.0
    3. Taylor expanded around inf to get
      \[\color{red}{\frac{1}{n \cdot x} - \left(\frac{\log x}{{n}^2 \cdot x} + \frac{1}{2} \cdot \frac{1}{n \cdot {x}^2}\right)} \leadsto \color{blue}{\frac{1}{n \cdot x} - \left(\frac{\log x}{{n}^2 \cdot x} + \frac{1}{2} \cdot \frac{1}{n \cdot {x}^2}\right)}\]
      16.0
    4. Applied taylor to get
      \[\frac{1}{n \cdot x} - \left(\frac{\log x}{{n}^2 \cdot x} + \frac{1}{2} \cdot \frac{1}{n \cdot {x}^2}\right) \leadsto \frac{1}{n \cdot x} - \left(\frac{1}{2} \cdot \frac{1}{n \cdot {x}^2} - \frac{\log x}{{n}^2 \cdot x}\right)\]
      2.8
    5. Taylor expanded around inf to get
      \[\frac{1}{n \cdot x} - \color{red}{\left(\frac{1}{2} \cdot \frac{1}{n \cdot {x}^2} - \frac{\log x}{{n}^2 \cdot x}\right)} \leadsto \frac{1}{n \cdot x} - \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{n \cdot {x}^2} - \frac{\log x}{{n}^2 \cdot x}\right)}\]
      2.8
    6. Applied simplify to get
      \[\frac{1}{n \cdot x} - \left(\frac{1}{2} \cdot \frac{1}{n \cdot {x}^2} - \frac{\log x}{{n}^2 \cdot x}\right) \leadsto \left(\frac{\log x}{\left(n \cdot x\right) \cdot n} + \frac{\frac{1}{n}}{x}\right) - \frac{\frac{1}{2}}{\left(x \cdot x\right) \cdot n}\]
      0.4
    7. Applied final simplification
    8. Applied simplify to get
      \[\color{red}{\left(\frac{\log x}{\left(n \cdot x\right) \cdot n} + \frac{\frac{1}{n}}{x}\right) - \frac{\frac{1}{2}}{\left(x \cdot x\right) \cdot n}} \leadsto \color{blue}{\frac{\log x}{x \cdot \left(n \cdot n\right)} + \left(\frac{1}{n \cdot x} - \frac{\frac{\frac{1}{2}}{n}}{{x}^2}\right)}\]
      2.7

    if 3.8668015f+18 < x

    1. Started with
      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
      5.8
    2. Using strategy rm
      5.8
    3. Applied add-cube-cbrt to get
      \[\color{red}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \leadsto \color{blue}{{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}^3}\]
      5.8
    4. Using strategy rm
      5.8
    5. Applied add-cube-cbrt to get
      \[{\color{red}{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}}^3 \leadsto {\color{blue}{\left({\left(\sqrt[3]{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)}^3\right)}}^3\]
      5.8
    6. Using strategy rm
      5.8
    7. Applied flip-- to get
      \[{\left({\left(\sqrt[3]{\sqrt[3]{\color{red}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}}\right)}^3\right)}^3 \leadsto {\left({\left(\sqrt[3]{\sqrt[3]{\color{blue}{\frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^2 - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^2}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}}}}\right)}^3\right)}^3\]
      15.9
    8. Applied cbrt-div to get
      \[{\left({\left(\sqrt[3]{\color{red}{\sqrt[3]{\frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^2 - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^2}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}}}}\right)}^3\right)}^3 \leadsto {\left({\left(\sqrt[3]{\color{blue}{\frac{\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^2 - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^2}}{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}}}}\right)}^3\right)}^3\]
      15.9
    9. Applied taylor to get
      \[{\left({\left(\sqrt[3]{\frac{\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^2 - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^2}}{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}}}\right)}^3\right)}^3 \leadsto {\left({\left(\sqrt[3]{\frac{\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^2 - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^2}}{\sqrt[3]{1 + {x}^{\left(\frac{1}{n}\right)}}}}\right)}^3\right)}^3\]
      5.8
    10. Taylor expanded around 0 to get
      \[{\left({\left(\sqrt[3]{\frac{\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^2 - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^2}}{\sqrt[3]{\color{red}{1} + {x}^{\left(\frac{1}{n}\right)}}}}\right)}^3\right)}^3 \leadsto {\left({\left(\sqrt[3]{\frac{\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^2 - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^2}}{\sqrt[3]{\color{blue}{1} + {x}^{\left(\frac{1}{n}\right)}}}}\right)}^3\right)}^3\]
      5.8
    11. Applied simplify to get
      \[{\left({\left(\sqrt[3]{\frac{\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^2 - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^2}}{\sqrt[3]{1 + {x}^{\left(\frac{1}{n}\right)}}}}\right)}^3\right)}^3 \leadsto \frac{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}\right)}{{x}^{\left(\frac{1}{n}\right)} + 1}\]
      5.8
    12. 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))))