\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
Test:
NMSE problem 3.4.6
Bits:
128 bits
Bits error versus x
Bits error versus n
Time: 43.8 s
Input Error: 38.5
Output Error: 18.0
Log: ⚲
Profile: 🕒
\(\begin{cases} \left(\left(\frac{1}{n \cdot x} + 1\right) - e^{\frac{\log x}{n}}\right) - \frac{\log x}{n} & \text{when } n \le -3.4933246215613803 \cdot 10^{+136} \\ \frac{\log x}{x \cdot \left(n \cdot n\right)} + \left(\frac{1}{n \cdot x} - \frac{\frac{\frac{1}{2}}{n}}{{x}^2}\right) & \text{when } n \le -523.3981319783818 \\ \log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right) & \text{when } n \le 3.985901124844516 \\ \frac{\log x}{x \cdot \left(n \cdot n\right)} + \left(\frac{1}{n \cdot x} - \frac{\frac{\frac{1}{2}}{n}}{{x}^2}\right) & \text{when } n \le 1.911985870044722 \cdot 10^{+190} \\ \left(\left(\frac{1}{n \cdot x} + 1\right) - e^{\frac{\log x}{n}}\right) - \frac{\log x}{n} & \text{when } n \le 1.8631211262211848 \cdot 10^{+264} \\ \frac{\log x}{x \cdot \left(n \cdot n\right)} + \left(\frac{1}{n \cdot x} - \frac{\frac{\frac{1}{2}}{n}}{{x}^2}\right) & \text{otherwise} \end{cases}\)

    if n < -3.4933246215613803e+136 or 1.911985870044722e+190 < n < 1.8631211262211848e+264

    1. Started with
      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
      60.7
    2. Applied taylor to get
      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leadsto \left(\left(1 + \frac{1}{n \cdot x}\right) - \frac{\log x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\]
      51.4
    3. Taylor expanded around inf to get
      \[\color{red}{\left(\left(1 + \frac{1}{n \cdot x}\right) - \frac{\log x}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leadsto \color{blue}{\left(\left(1 + \frac{1}{n \cdot x}\right) - \frac{\log x}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
      51.4
    4. Applied taylor to get
      \[\left(\left(1 + \frac{1}{n \cdot x}\right) - \frac{\log x}{n}\right) - {x}^{\left(\frac{1}{n}\right)} \leadsto \left(\left(1 + \frac{1}{n \cdot x}\right) - \frac{\log x}{n}\right) - e^{\frac{\log x}{n}}\]
      51.4
    5. Taylor expanded around 0 to get
      \[\left(\left(1 + \frac{1}{n \cdot x}\right) - \frac{\log x}{n}\right) - \color{red}{e^{\frac{\log x}{n}}} \leadsto \left(\left(1 + \frac{1}{n \cdot x}\right) - \frac{\log x}{n}\right) - \color{blue}{e^{\frac{\log x}{n}}}\]
      51.4
    6. Applied simplify to get
      \[\left(\left(1 + \frac{1}{n \cdot x}\right) - \frac{\log x}{n}\right) - e^{\frac{\log x}{n}} \leadsto \left(\left(\frac{1}{n \cdot x} + 1\right) - e^{\frac{\log x}{n}}\right) - \frac{\log x}{n}\]
      47.1
    7. Applied final simplification

    if -3.4933246215613803e+136 < n < -523.3981319783818 or 3.985901124844516 < n < 1.911985870044722e+190 or 1.8631211262211848e+264 < n

    1. Started with
      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
      49.0
    2. Applied taylor to get
      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leadsto \frac{1}{n \cdot x} - \left(\frac{\log x}{{n}^2 \cdot x} + \frac{1}{2} \cdot \frac{1}{n \cdot {x}^2}\right)\]
      8.9
    3. Taylor expanded around inf to get
      \[\color{red}{\frac{1}{n \cdot x} - \left(\frac{\log x}{{n}^2 \cdot x} + \frac{1}{2} \cdot \frac{1}{n \cdot {x}^2}\right)} \leadsto \color{blue}{\frac{1}{n \cdot x} - \left(\frac{\log x}{{n}^2 \cdot x} + \frac{1}{2} \cdot \frac{1}{n \cdot {x}^2}\right)}\]
      8.9
    4. Applied taylor to get
      \[\frac{1}{n \cdot x} - \left(\frac{\log x}{{n}^2 \cdot x} + \frac{1}{2} \cdot \frac{1}{n \cdot {x}^2}\right) \leadsto \frac{1}{n \cdot x} - \left(\frac{1}{2} \cdot \frac{1}{n \cdot {x}^2} - \frac{\log x}{{n}^2 \cdot x}\right)\]
      7.4
    5. Taylor expanded around inf to get
      \[\frac{1}{n \cdot x} - \color{red}{\left(\frac{1}{2} \cdot \frac{1}{n \cdot {x}^2} - \frac{\log x}{{n}^2 \cdot x}\right)} \leadsto \frac{1}{n \cdot x} - \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{n \cdot {x}^2} - \frac{\log x}{{n}^2 \cdot x}\right)}\]
      7.4
    6. Applied simplify to get
      \[\frac{1}{n \cdot x} - \left(\frac{1}{2} \cdot \frac{1}{n \cdot {x}^2} - \frac{\log x}{{n}^2 \cdot x}\right) \leadsto \left(\frac{\log x}{\left(n \cdot x\right) \cdot n} + \frac{\frac{1}{n}}{x}\right) - \frac{\frac{1}{2}}{\left(x \cdot x\right) \cdot n}\]
      6.7
    7. Applied final simplification
    8. Applied simplify to get
      \[\color{red}{\left(\frac{\log x}{\left(n \cdot x\right) \cdot n} + \frac{\frac{1}{n}}{x}\right) - \frac{\frac{1}{2}}{\left(x \cdot x\right) \cdot n}} \leadsto \color{blue}{\frac{\log x}{x \cdot \left(n \cdot n\right)} + \left(\frac{1}{n \cdot x} - \frac{\frac{\frac{1}{2}}{n}}{{x}^2}\right)}\]
      7.4

    if -523.3981319783818 < n < 3.985901124844516

    1. Started with
      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
      1.4
    2. Using strategy rm
      1.4
    3. Applied add-log-exp to get
      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \color{red}{{x}^{\left(\frac{1}{n}\right)}} \leadsto {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \color{blue}{\log \left(e^{{x}^{\left(\frac{1}{n}\right)}}\right)}\]
      1.5
    4. Applied add-log-exp to get
      \[\color{red}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - \log \left(e^{{x}^{\left(\frac{1}{n}\right)}}\right) \leadsto \color{blue}{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right)} - \log \left(e^{{x}^{\left(\frac{1}{n}\right)}}\right)\]
      1.5
    5. Applied diff-log to get
      \[\color{red}{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right) - \log \left(e^{{x}^{\left(\frac{1}{n}\right)}}\right)} \leadsto \color{blue}{\log \left(\frac{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}}{e^{{x}^{\left(\frac{1}{n}\right)}}}\right)}\]
      1.5
    6. Applied simplify to get
      \[\log \color{red}{\left(\frac{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}}{e^{{x}^{\left(\frac{1}{n}\right)}}}\right)} \leadsto \log \color{blue}{\left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}\]
      1.5
  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))))