\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
Test:
NMSE problem 3.4.6
Bits:
128 bits
Bits error versus x
Bits error versus n
Time: 33.7 s
Input Error: 15.7
Output Error: 10.6
Log: ⚲
Profile: 🕒
\(\begin{cases} e^{\frac{x - \left(\frac{1}{2} - x \cdot \frac{1}{3}\right) \cdot \left(x \cdot x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)} & \text{when } x \le 7.661514f-12 \\ \left(\left(\frac{1}{n \cdot x} + 1\right) - e^{\frac{\log x}{n}}\right) - \frac{\log x}{n} & \text{when } x \le 0.8876801f0 \\ \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\frac{\frac{\frac{1}{2}}{x}}{n} - \left(\frac{\log x}{x} \cdot \frac{\frac{\frac{-1}{4}}{n}}{n} + \frac{\frac{\frac{1}{4}}{n}}{{x}^2}\right)\right) & \text{otherwise} \end{cases}\)

    if x < 7.661514f-12

    1. Started with
      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
      19.3
    2. Using strategy rm
      19.3
    3. Applied add-exp-log to get
      \[{\color{red}{\left(x + 1\right)}}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leadsto {\color{blue}{\left(e^{\log \left(x + 1\right)}\right)}}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
      19.3
    4. Applied pow-exp to get
      \[\color{red}{{\left(e^{\log \left(x + 1\right)}\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{1}{n}\right)} \leadsto \color{blue}{e^{\log \left(x + 1\right) \cdot \frac{1}{n}}} - {x}^{\left(\frac{1}{n}\right)}\]
      19.3
    5. Applied simplify to get
      \[e^{\color{red}{\log \left(x + 1\right) \cdot \frac{1}{n}}} - {x}^{\left(\frac{1}{n}\right)} \leadsto e^{\color{blue}{\frac{\log \left(x + 1\right)}{n}}} - {x}^{\left(\frac{1}{n}\right)}\]
      19.3
    6. Applied taylor to get
      \[e^{\frac{\log \left(x + 1\right)}{n}} - {x}^{\left(\frac{1}{n}\right)} \leadsto e^{\frac{\left(\frac{1}{3} \cdot {x}^{3} + x\right) - \frac{1}{2} \cdot {x}^2}{n}} - {x}^{\left(\frac{1}{n}\right)}\]
      17.9
    7. Taylor expanded around 0 to get
      \[e^{\frac{\color{red}{\left(\frac{1}{3} \cdot {x}^{3} + x\right) - \frac{1}{2} \cdot {x}^2}}{n}} - {x}^{\left(\frac{1}{n}\right)} \leadsto e^{\frac{\color{blue}{\left(\frac{1}{3} \cdot {x}^{3} + x\right) - \frac{1}{2} \cdot {x}^2}}{n}} - {x}^{\left(\frac{1}{n}\right)}\]
      17.9
    8. Applied simplify to get
      \[e^{\frac{\left(\frac{1}{3} \cdot {x}^{3} + x\right) - \frac{1}{2} \cdot {x}^2}{n}} - {x}^{\left(\frac{1}{n}\right)} \leadsto e^{\frac{\left(\frac{1}{3} \cdot {x}^3 + x\right) - \frac{1}{2} \cdot \left(x \cdot x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\]
      17.9
    9. Applied final simplification
    10. Applied simplify to get
      \[\color{red}{e^{\frac{\left(\frac{1}{3} \cdot {x}^3 + x\right) - \frac{1}{2} \cdot \left(x \cdot x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}} \leadsto \color{blue}{e^{\frac{x - \left(\frac{1}{2} - x \cdot \frac{1}{3}\right) \cdot \left(x \cdot x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}}\]
      17.9

    if 7.661514f-12 < x < 0.8876801f0

    1. Started with
      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
      28.1
    2. Applied taylor to get
      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leadsto \left(\left(1 + \frac{1}{n \cdot x}\right) - \frac{\log x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\]
      14.6
    3. Taylor expanded around inf to get
      \[\color{red}{\left(\left(1 + \frac{1}{n \cdot x}\right) - \frac{\log x}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leadsto \color{blue}{\left(\left(1 + \frac{1}{n \cdot x}\right) - \frac{\log x}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
      14.6
    4. Applied taylor to get
      \[\left(\left(1 + \frac{1}{n \cdot x}\right) - \frac{\log x}{n}\right) - {x}^{\left(\frac{1}{n}\right)} \leadsto \left(\left(1 + \frac{1}{n \cdot x}\right) - \frac{\log x}{n}\right) - e^{\frac{\log x}{n}}\]
      14.6
    5. Taylor expanded around 0 to get
      \[\left(\left(1 + \frac{1}{n \cdot x}\right) - \frac{\log x}{n}\right) - \color{red}{e^{\frac{\log x}{n}}} \leadsto \left(\left(1 + \frac{1}{n \cdot x}\right) - \frac{\log x}{n}\right) - \color{blue}{e^{\frac{\log x}{n}}}\]
      14.6
    6. Applied simplify to get
      \[\left(\left(1 + \frac{1}{n \cdot x}\right) - \frac{\log x}{n}\right) - e^{\frac{\log x}{n}} \leadsto \left(\left(\frac{1}{n \cdot x} + 1\right) - e^{\frac{\log x}{n}}\right) - \frac{\log x}{n}\]
      13.8
    7. Applied final simplification

    if 0.8876801f0 < x

    1. Started with
      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
      10.5
    2. Using strategy rm
      10.5
    3. Applied add-sqr-sqrt to get
      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \color{red}{{x}^{\left(\frac{1}{n}\right)}} \leadsto {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \color{blue}{{\left(\sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)}^2}\]
      11.0
    4. Applied add-sqr-sqrt to get
      \[\color{red}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - {\left(\sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)}^2 \leadsto \color{blue}{{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right)}^2} - {\left(\sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)}^2\]
      10.5
    5. Applied difference-of-squares to get
      \[\color{red}{{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right)}^2 - {\left(\sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)}^2} \leadsto \color{blue}{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)}\]
      10.5
    6. Applied taylor to get
      \[\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \leadsto \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\frac{1}{2} \cdot \frac{1}{n \cdot x} - \left(\frac{1}{4} \cdot \frac{\log x}{{n}^2 \cdot x} + \frac{1}{4} \cdot \frac{1}{n \cdot {x}^2}\right)\right)\]
      7.8
    7. Taylor expanded around inf to get
      \[\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \color{red}{\left(\frac{1}{2} \cdot \frac{1}{n \cdot x} - \left(\frac{1}{4} \cdot \frac{\log x}{{n}^2 \cdot x} + \frac{1}{4} \cdot \frac{1}{n \cdot {x}^2}\right)\right)} \leadsto \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{n \cdot x} - \left(\frac{1}{4} \cdot \frac{\log x}{{n}^2 \cdot x} + \frac{1}{4} \cdot \frac{1}{n \cdot {x}^2}\right)\right)}\]
      7.8
    8. Applied simplify to get
      \[\color{red}{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\frac{1}{2} \cdot \frac{1}{n \cdot x} - \left(\frac{1}{4} \cdot \frac{\log x}{{n}^2 \cdot x} + \frac{1}{4} \cdot \frac{1}{n \cdot {x}^2}\right)\right)} \leadsto \color{blue}{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\left(\frac{\frac{\frac{1}{2}}{n}}{x} - \frac{\frac{\log x}{\frac{x}{\frac{1}{4}}}}{n \cdot n}\right) - \frac{\frac{\frac{1}{4}}{n}}{{x}^2}\right)}\]
      6.7
    9. Applied taylor to get
      \[\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\left(\frac{\frac{\frac{1}{2}}{n}}{x} - \frac{\frac{\log x}{\frac{x}{\frac{1}{4}}}}{n \cdot n}\right) - \frac{\frac{\frac{1}{4}}{n}}{{x}^2}\right) \leadsto \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\left(\frac{\frac{\frac{1}{2}}{n}}{x} - \frac{-1}{4} \cdot \frac{\log x}{{n}^2 \cdot x}\right) - \frac{\frac{\frac{1}{4}}{n}}{{x}^2}\right)\]
      4.8
    10. Taylor expanded around inf to get
      \[\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\left(\frac{\frac{\frac{1}{2}}{n}}{x} - \color{red}{\frac{-1}{4} \cdot \frac{\log x}{{n}^2 \cdot x}}\right) - \frac{\frac{\frac{1}{4}}{n}}{{x}^2}\right) \leadsto \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\left(\frac{\frac{\frac{1}{2}}{n}}{x} - \color{blue}{\frac{-1}{4} \cdot \frac{\log x}{{n}^2 \cdot x}}\right) - \frac{\frac{\frac{1}{4}}{n}}{{x}^2}\right)\]
      4.8
    11. Applied simplify to get
      \[\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\left(\frac{\frac{\frac{1}{2}}{n}}{x} - \frac{-1}{4} \cdot \frac{\log x}{{n}^2 \cdot x}\right) - \frac{\frac{\frac{1}{4}}{n}}{{x}^2}\right) \leadsto \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\frac{\frac{\frac{1}{2}}{x}}{n} - \left(\frac{\log x}{x} \cdot \frac{\frac{\frac{-1}{4}}{n}}{n} + \frac{\frac{\frac{1}{4}}{n}}{{x}^2}\right)\right)\]
      5.3
    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))))