\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
Test:
NMSE problem 3.4.6
Bits:
128 bits
Bits error versus x
Bits error versus n
Time: 24.4 s
Input Error: 19.2
Output Error: 17.3
Log: ⚲
Profile: 🕒
\(\begin{cases} (\left(\log_* (1 + \frac{1}{x}) \cdot \frac{1}{n}\right) * \left((\left(\frac{\log_* (1 + \frac{1}{x})}{n}\right) * \frac{1}{2} + 1)_*\right) + \left(1 - {x}^{\left(\frac{1}{n}\right)}\right))_* & \text{when } n \le -3.3318707f+08 \\ e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)} & \text{when } n \le 2.3951968f+08 \\ (\left(\frac{\log_* (1 + \frac{1}{x})}{n}\right) * \left((\left(\frac{\log_* (1 + \frac{1}{x})}{n}\right) * \frac{1}{2} + 1)_*\right) + \left(1 - {x}^{\left(\frac{1}{n}\right)}\right))_* & \text{otherwise} \end{cases}\)

    if n < -3.3318707f+08

    1. Started with
      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
      28.5
    2. Using strategy rm
      28.5
    3. Applied add-exp-log to get
      \[{\color{red}{\left(x + 1\right)}}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leadsto {\color{blue}{\left(e^{\log \left(x + 1\right)}\right)}}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
      28.5
    4. Applied pow-exp to get
      \[\color{red}{{\left(e^{\log \left(x + 1\right)}\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{1}{n}\right)} \leadsto \color{blue}{e^{\log \left(x + 1\right) \cdot \frac{1}{n}}} - {x}^{\left(\frac{1}{n}\right)}\]
      28.5
    5. Applied simplify to get
      \[e^{\color{red}{\log \left(x + 1\right) \cdot \frac{1}{n}}} - {x}^{\left(\frac{1}{n}\right)} \leadsto e^{\color{blue}{\frac{\log_* (1 + x)}{n}}} - {x}^{\left(\frac{1}{n}\right)}\]
      28.5
    6. Applied taylor to get
      \[e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)} \leadsto \left(\frac{\log_* (1 + \frac{1}{x})}{n} + \left(1 + \frac{1}{2} \cdot \frac{{\left(\log_* (1 + \frac{1}{x})\right)}^2}{{n}^2}\right)\right) - {x}^{\left(\frac{1}{n}\right)}\]
      27.3
    7. Taylor expanded around inf to get
      \[\color{red}{\left(\frac{\log_* (1 + \frac{1}{x})}{n} + \left(1 + \frac{1}{2} \cdot \frac{{\left(\log_* (1 + \frac{1}{x})\right)}^2}{{n}^2}\right)\right)} - {x}^{\left(\frac{1}{n}\right)} \leadsto \color{blue}{\left(\frac{\log_* (1 + \frac{1}{x})}{n} + \left(1 + \frac{1}{2} \cdot \frac{{\left(\log_* (1 + \frac{1}{x})\right)}^2}{{n}^2}\right)\right)} - {x}^{\left(\frac{1}{n}\right)}\]
      27.3
    8. Applied simplify to get
      \[\color{red}{\left(\frac{\log_* (1 + \frac{1}{x})}{n} + \left(1 + \frac{1}{2} \cdot \frac{{\left(\log_* (1 + \frac{1}{x})\right)}^2}{{n}^2}\right)\right) - {x}^{\left(\frac{1}{n}\right)}} \leadsto \color{blue}{(\left(\frac{\log_* (1 + \frac{1}{x})}{n}\right) * \left((\left(\frac{\log_* (1 + \frac{1}{x})}{n}\right) * \frac{1}{2} + 1)_*\right) + \left(1 - {x}^{\left(\frac{1}{n}\right)}\right))_*}\]
      25.7
    9. Using strategy rm
      25.7
    10. Applied div-inv to get
      \[(\color{red}{\left(\frac{\log_* (1 + \frac{1}{x})}{n}\right)} * \left((\left(\frac{\log_* (1 + \frac{1}{x})}{n}\right) * \frac{1}{2} + 1)_*\right) + \left(1 - {x}^{\left(\frac{1}{n}\right)}\right))_* \leadsto (\color{blue}{\left(\log_* (1 + \frac{1}{x}) \cdot \frac{1}{n}\right)} * \left((\left(\frac{\log_* (1 + \frac{1}{x})}{n}\right) * \frac{1}{2} + 1)_*\right) + \left(1 - {x}^{\left(\frac{1}{n}\right)}\right))_*\]
      25.7

    if -3.3318707f+08 < n < 2.3951968f+08

    1. Started with
      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
      7.5
    2. Using strategy rm
      7.5
    3. Applied add-exp-log to get
      \[{\color{red}{\left(x + 1\right)}}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leadsto {\color{blue}{\left(e^{\log \left(x + 1\right)}\right)}}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
      7.7
    4. Applied pow-exp to get
      \[\color{red}{{\left(e^{\log \left(x + 1\right)}\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{1}{n}\right)} \leadsto \color{blue}{e^{\log \left(x + 1\right) \cdot \frac{1}{n}}} - {x}^{\left(\frac{1}{n}\right)}\]
      7.7
    5. Applied simplify to get
      \[e^{\color{red}{\log \left(x + 1\right) \cdot \frac{1}{n}}} - {x}^{\left(\frac{1}{n}\right)} \leadsto e^{\color{blue}{\frac{\log_* (1 + x)}{n}}} - {x}^{\left(\frac{1}{n}\right)}\]
      6.5

    if 2.3951968f+08 < n

    1. Started with
      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
      28.3
    2. Using strategy rm
      28.3
    3. Applied add-exp-log to get
      \[{\color{red}{\left(x + 1\right)}}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leadsto {\color{blue}{\left(e^{\log \left(x + 1\right)}\right)}}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
      28.3
    4. Applied pow-exp to get
      \[\color{red}{{\left(e^{\log \left(x + 1\right)}\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{1}{n}\right)} \leadsto \color{blue}{e^{\log \left(x + 1\right) \cdot \frac{1}{n}}} - {x}^{\left(\frac{1}{n}\right)}\]
      28.3
    5. Applied simplify to get
      \[e^{\color{red}{\log \left(x + 1\right) \cdot \frac{1}{n}}} - {x}^{\left(\frac{1}{n}\right)} \leadsto e^{\color{blue}{\frac{\log_* (1 + x)}{n}}} - {x}^{\left(\frac{1}{n}\right)}\]
      28.3
    6. Applied taylor to get
      \[e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)} \leadsto \left(\frac{\log_* (1 + \frac{1}{x})}{n} + \left(1 + \frac{1}{2} \cdot \frac{{\left(\log_* (1 + \frac{1}{x})\right)}^2}{{n}^2}\right)\right) - {x}^{\left(\frac{1}{n}\right)}\]
      27.4
    7. Taylor expanded around inf to get
      \[\color{red}{\left(\frac{\log_* (1 + \frac{1}{x})}{n} + \left(1 + \frac{1}{2} \cdot \frac{{\left(\log_* (1 + \frac{1}{x})\right)}^2}{{n}^2}\right)\right)} - {x}^{\left(\frac{1}{n}\right)} \leadsto \color{blue}{\left(\frac{\log_* (1 + \frac{1}{x})}{n} + \left(1 + \frac{1}{2} \cdot \frac{{\left(\log_* (1 + \frac{1}{x})\right)}^2}{{n}^2}\right)\right)} - {x}^{\left(\frac{1}{n}\right)}\]
      27.4
    8. Applied simplify to get
      \[\color{red}{\left(\frac{\log_* (1 + \frac{1}{x})}{n} + \left(1 + \frac{1}{2} \cdot \frac{{\left(\log_* (1 + \frac{1}{x})\right)}^2}{{n}^2}\right)\right) - {x}^{\left(\frac{1}{n}\right)}} \leadsto \color{blue}{(\left(\frac{\log_* (1 + \frac{1}{x})}{n}\right) * \left((\left(\frac{\log_* (1 + \frac{1}{x})}{n}\right) * \frac{1}{2} + 1)_*\right) + \left(1 - {x}^{\left(\frac{1}{n}\right)}\right))_*}\]
      25.7
  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))))