\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
Test:
NMSE problem 3.4.6
Bits:
128 bits
Bits error versus x
Bits error versus n
Time: 53.5 s
Input Error: 30.7
Output Error: 7.9
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{\left(\frac{\frac{2}{x}}{n} - \frac{4}{x} \cdot \frac{\log x}{n \cdot n}\right) - \frac{\frac{1}{n}}{x \cdot x}}{{x}^{\left(\frac{1}{n}\right)} + {\left(1 + x\right)}^{\left(\frac{1}{n}\right)}} & \text{when } n \le -846584004.9676423 \\ \log \left({\left(\sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)}^3\right) & \text{when } n \le 1.0094449940811532 \cdot 10^{+18} \\ \frac{\left(\frac{\frac{2}{x}}{n} - \frac{4}{x} \cdot \frac{\log x}{n \cdot n}\right) - \frac{\frac{1}{n}}{x \cdot x}}{{x}^{\left(\frac{1}{n}\right)} + {\left(1 + x\right)}^{\left(\frac{1}{n}\right)}} & \text{otherwise} \end{cases}\)

    if n < -846584004.9676423

    1. Started with
      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
      44.4
    2. Using strategy rm
      44.4
    3. Applied add-log-exp to get
      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \color{red}{{x}^{\left(\frac{1}{n}\right)}} \leadsto {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \color{blue}{\log \left(e^{{x}^{\left(\frac{1}{n}\right)}}\right)}\]
      44.4
    4. Applied add-log-exp to get
      \[\color{red}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - \log \left(e^{{x}^{\left(\frac{1}{n}\right)}}\right) \leadsto \color{blue}{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right)} - \log \left(e^{{x}^{\left(\frac{1}{n}\right)}}\right)\]
      44.4
    5. Applied diff-log to get
      \[\color{red}{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right) - \log \left(e^{{x}^{\left(\frac{1}{n}\right)}}\right)} \leadsto \color{blue}{\log \left(\frac{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}}{e^{{x}^{\left(\frac{1}{n}\right)}}}\right)}\]
      44.4
    6. Applied simplify to get
      \[\log \color{red}{\left(\frac{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}}{e^{{x}^{\left(\frac{1}{n}\right)}}}\right)} \leadsto \log \color{blue}{\left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}\]
      44.4
    7. Using strategy rm
      44.4
    8. Applied flip-- to get
      \[\log \left(e^{\color{red}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right) \leadsto \log \left(e^{\color{blue}{\frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^2 - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^2}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}}}\right)\]
      44.4
    9. Applied taylor to get
      \[\log \left(e^{\frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^2 - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^2}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}}\right) \leadsto \log \left(e^{\frac{2 \cdot \frac{1}{n \cdot x} - \left(4 \cdot \frac{\log x}{{n}^2 \cdot x} + \frac{1}{n \cdot {x}^2}\right)}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}}\right)\]
      43.3
    10. Taylor expanded around inf to get
      \[\log \left(e^{\frac{\color{red}{2 \cdot \frac{1}{n \cdot x} - \left(4 \cdot \frac{\log x}{{n}^2 \cdot x} + \frac{1}{n \cdot {x}^2}\right)}}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}}\right) \leadsto \log \left(e^{\frac{\color{blue}{2 \cdot \frac{1}{n \cdot x} - \left(4 \cdot \frac{\log x}{{n}^2 \cdot x} + \frac{1}{n \cdot {x}^2}\right)}}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}}\right)\]
      43.3
    11. Applied simplify to get
      \[\log \left(e^{\frac{2 \cdot \frac{1}{n \cdot x} - \left(4 \cdot \frac{\log x}{{n}^2 \cdot x} + \frac{1}{n \cdot {x}^2}\right)}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}}\right) \leadsto \frac{\left(\frac{\frac{2}{x}}{n} - \frac{4}{x} \cdot \frac{\log x}{n \cdot n}\right) - \frac{\frac{1}{n}}{x \cdot x}}{{x}^{\left(\frac{1}{n}\right)} + {\left(1 + x\right)}^{\left(\frac{1}{n}\right)}}\]
      8.6
    12. Applied final simplification

    if -846584004.9676423 < n < 1.0094449940811532e+18

    1. Started with
      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
      5.1
    2. Using strategy rm
      5.1
    3. Applied add-log-exp to get
      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \color{red}{{x}^{\left(\frac{1}{n}\right)}} \leadsto {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \color{blue}{\log \left(e^{{x}^{\left(\frac{1}{n}\right)}}\right)}\]
      5.2
    4. Applied add-log-exp to get
      \[\color{red}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - \log \left(e^{{x}^{\left(\frac{1}{n}\right)}}\right) \leadsto \color{blue}{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right)} - \log \left(e^{{x}^{\left(\frac{1}{n}\right)}}\right)\]
      5.1
    5. Applied diff-log to get
      \[\color{red}{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right) - \log \left(e^{{x}^{\left(\frac{1}{n}\right)}}\right)} \leadsto \color{blue}{\log \left(\frac{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}}{e^{{x}^{\left(\frac{1}{n}\right)}}}\right)}\]
      5.1
    6. Applied simplify to get
      \[\log \color{red}{\left(\frac{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}}{e^{{x}^{\left(\frac{1}{n}\right)}}}\right)} \leadsto \log \color{blue}{\left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}\]
      5.1
    7. Using strategy rm
      5.1
    8. Applied add-cube-cbrt to get
      \[\log \color{red}{\left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)} \leadsto \log \color{blue}{\left({\left(\sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)}^3\right)}\]
      5.8

    if 1.0094449940811532e+18 < n

    1. Started with
      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
      43.1
    2. Using strategy rm
      43.1
    3. Applied add-log-exp to get
      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \color{red}{{x}^{\left(\frac{1}{n}\right)}} \leadsto {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \color{blue}{\log \left(e^{{x}^{\left(\frac{1}{n}\right)}}\right)}\]
      43.1
    4. Applied add-log-exp to get
      \[\color{red}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - \log \left(e^{{x}^{\left(\frac{1}{n}\right)}}\right) \leadsto \color{blue}{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right)} - \log \left(e^{{x}^{\left(\frac{1}{n}\right)}}\right)\]
      43.1
    5. Applied diff-log to get
      \[\color{red}{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right) - \log \left(e^{{x}^{\left(\frac{1}{n}\right)}}\right)} \leadsto \color{blue}{\log \left(\frac{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}}{e^{{x}^{\left(\frac{1}{n}\right)}}}\right)}\]
      43.1
    6. Applied simplify to get
      \[\log \color{red}{\left(\frac{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}}{e^{{x}^{\left(\frac{1}{n}\right)}}}\right)} \leadsto \log \color{blue}{\left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}\]
      43.1
    7. Using strategy rm
      43.1
    8. Applied flip-- to get
      \[\log \left(e^{\color{red}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right) \leadsto \log \left(e^{\color{blue}{\frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^2 - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^2}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}}}\right)\]
      43.1
    9. Applied taylor to get
      \[\log \left(e^{\frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^2 - {\left({x}^{\left(\frac{1}{n}\right)}\right)}^2}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}}\right) \leadsto \log \left(e^{\frac{2 \cdot \frac{1}{n \cdot x} - \left(4 \cdot \frac{\log x}{{n}^2 \cdot x} + \frac{1}{n \cdot {x}^2}\right)}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}}\right)\]
      41.8
    10. Taylor expanded around inf to get
      \[\log \left(e^{\frac{\color{red}{2 \cdot \frac{1}{n \cdot x} - \left(4 \cdot \frac{\log x}{{n}^2 \cdot x} + \frac{1}{n \cdot {x}^2}\right)}}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}}\right) \leadsto \log \left(e^{\frac{\color{blue}{2 \cdot \frac{1}{n \cdot x} - \left(4 \cdot \frac{\log x}{{n}^2 \cdot x} + \frac{1}{n \cdot {x}^2}\right)}}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}}\right)\]
      41.8
    11. Applied simplify to get
      \[\log \left(e^{\frac{2 \cdot \frac{1}{n \cdot x} - \left(4 \cdot \frac{\log x}{{n}^2 \cdot x} + \frac{1}{n \cdot {x}^2}\right)}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}}\right) \leadsto \frac{\left(\frac{\frac{2}{x}}{n} - \frac{4}{x} \cdot \frac{\log x}{n \cdot n}\right) - \frac{\frac{1}{n}}{x \cdot x}}{{x}^{\left(\frac{1}{n}\right)} + {\left(1 + x\right)}^{\left(\frac{1}{n}\right)}}\]
      9.2
    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))))