\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
Test:
NMSE problem 3.4.6
Bits:
128 bits
Bits error versus x
Bits error versus n
Time: 47.4 s
Input Error: 32.6
Output Error: 28.0
Log: ⚲
Profile: 🕒
\(\begin{cases} e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)} & \text{when } x \le 1.751947898743096 \cdot 10^{-155} \\ e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)} & \text{when } x \le 4.2915703207789075 \\ \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\frac{\frac{\frac{1}{2}}{x}}{n} - (\left(\frac{\frac{1}{4}}{n \cdot n}\right) * \left(\frac{\log x}{x}\right) + \left(\frac{\frac{1}{4}}{n \cdot {x}^2}\right))_*\right) & \text{when } x \le 3.0165602320725886 \cdot 10^{+73} \\ e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)} & \text{when } x \le 8.7651298832822 \cdot 10^{+100} \\ \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\frac{\frac{\frac{1}{2}}{x}}{n} - (\left(\frac{\frac{1}{4}}{n \cdot n}\right) * \left(\frac{\log x}{x}\right) + \left(\frac{\frac{1}{4}}{n \cdot {x}^2}\right))_*\right) & \text{when } x \le 2.1269679020085675 \cdot 10^{+145} \\ e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)} & \text{when } x \le 3.1044253479896454 \cdot 10^{+201} \\ \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\frac{\frac{\frac{1}{2}}{x}}{n} - (\left(\frac{\frac{1}{4}}{n \cdot n}\right) * \left(\frac{\log x}{x}\right) + \left(\frac{\frac{1}{4}}{n \cdot {x}^2}\right))_*\right) & \text{when } x \le 1.5820368822243981 \cdot 10^{+217} \\ e^{\frac{\log_* (1 + x)}{n}} - {x}^{\left(\frac{1}{n}\right)} & \text{otherwise} \end{cases}\)

    if x < 1.751947898743096e-155

    1. Started with
      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
      42.6
    2. Using strategy rm
      42.6
    3. Applied add-exp-log to get
      \[{\color{red}{\left(x + 1\right)}}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leadsto {\color{blue}{\left(e^{\log \left(x + 1\right)}\right)}}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
      42.6
    4. Applied pow-exp to get
      \[\color{red}{{\left(e^{\log \left(x + 1\right)}\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{1}{n}\right)} \leadsto \color{blue}{e^{\log \left(x + 1\right) \cdot \frac{1}{n}}} - {x}^{\left(\frac{1}{n}\right)}\]
      42.6
    5. Applied simplify to get
      \[e^{\color{red}{\log \left(x + 1\right) \cdot \frac{1}{n}}} - {x}^{\left(\frac{1}{n}\right)} \leadsto e^{\color{blue}{\frac{\log_* (1 + x)}{n}}} - {x}^{\left(\frac{1}{n}\right)}\]
      41.9

    if 1.751947898743096e-155 < x < 4.2915703207789075

    1. Started with
      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
      52.6
    2. Using strategy rm
      52.6
    3. Applied add-exp-log to get
      \[{\color{red}{\left(x + 1\right)}}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leadsto {\color{blue}{\left(e^{\log \left(x + 1\right)}\right)}}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
      52.6
    4. Applied pow-exp to get
      \[\color{red}{{\left(e^{\log \left(x + 1\right)}\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{1}{n}\right)} \leadsto \color{blue}{e^{\log \left(x + 1\right) \cdot \frac{1}{n}}} - {x}^{\left(\frac{1}{n}\right)}\]
      52.6
    5. Applied simplify to get
      \[e^{\color{red}{\log \left(x + 1\right) \cdot \frac{1}{n}}} - {x}^{\left(\frac{1}{n}\right)} \leadsto e^{\color{blue}{\frac{\log_* (1 + x)}{n}}} - {x}^{\left(\frac{1}{n}\right)}\]
      52.0

    if 4.2915703207789075 < x < 3.0165602320725886e+73 or 8.7651298832822e+100 < x < 2.1269679020085675e+145 or 3.1044253479896454e+201 < x < 1.5820368822243981e+217

    1. Started with
      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
      29.1
    2. Using strategy rm
      29.1
    3. Applied add-cube-cbrt to get
      \[\color{red}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \leadsto \color{blue}{{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}^3}\]
      29.1
    4. Using strategy rm
      29.1
    5. Applied add-sqr-sqrt to get
      \[{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \color{red}{{x}^{\left(\frac{1}{n}\right)}}}\right)}^3 \leadsto {\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \color{blue}{{\left(\sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)}^2}}\right)}^3\]
      29.0
    6. Applied add-sqr-sqrt to get
      \[{\left(\sqrt[3]{\color{red}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - {\left(\sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)}^2}\right)}^3 \leadsto {\left(\sqrt[3]{\color{blue}{{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right)}^2} - {\left(\sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)}^2}\right)}^3\]
      29.1
    7. Applied difference-of-squares to get
      \[{\left(\sqrt[3]{\color{red}{{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right)}^2 - {\left(\sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)}^2}}\right)}^3 \leadsto {\left(\sqrt[3]{\color{blue}{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)}}\right)}^3\]
      29.1
    8. Applied cbrt-prod to get
      \[{\color{red}{\left(\sqrt[3]{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)}\right)}}^3 \leadsto {\color{blue}{\left(\sqrt[3]{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}} \cdot \sqrt[3]{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}}\right)}}^3\]
      29.1
    9. Applied taylor to get
      \[{\left(\sqrt[3]{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}} \cdot \sqrt[3]{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}}\right)}^3 \leadsto {\left(\sqrt[3]{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}} \cdot \sqrt[3]{\frac{1}{2} \cdot \frac{1}{n \cdot x} - \left(\frac{1}{4} \cdot \frac{\log x}{{n}^2 \cdot x} + \frac{1}{4} \cdot \frac{1}{n \cdot {x}^2}\right)}\right)}^3\]
      12.1
    10. Taylor expanded around inf to get
      \[{\left(\sqrt[3]{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}} \cdot \sqrt[3]{\color{red}{\frac{1}{2} \cdot \frac{1}{n \cdot x} - \left(\frac{1}{4} \cdot \frac{\log x}{{n}^2 \cdot x} + \frac{1}{4} \cdot \frac{1}{n \cdot {x}^2}\right)}}\right)}^3 \leadsto {\left(\sqrt[3]{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}} \cdot \sqrt[3]{\color{blue}{\frac{1}{2} \cdot \frac{1}{n \cdot x} - \left(\frac{1}{4} \cdot \frac{\log x}{{n}^2 \cdot x} + \frac{1}{4} \cdot \frac{1}{n \cdot {x}^2}\right)}}\right)}^3\]
      12.1
    11. Applied simplify to get
      \[{\left(\sqrt[3]{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}} \cdot \sqrt[3]{\frac{1}{2} \cdot \frac{1}{n \cdot x} - \left(\frac{1}{4} \cdot \frac{\log x}{{n}^2 \cdot x} + \frac{1}{4} \cdot \frac{1}{n \cdot {x}^2}\right)}\right)}^3 \leadsto \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\frac{\frac{\frac{1}{2}}{x}}{n} - (\left(\frac{\frac{1}{4}}{n \cdot n}\right) * \left(\frac{\log x}{x}\right) + \left(\frac{\frac{1}{4}}{n \cdot {x}^2}\right))_*\right)\]
      11.3
    12. Applied final simplification

    if 3.0165602320725886e+73 < x < 8.7651298832822e+100 or 2.1269679020085675e+145 < x < 3.1044253479896454e+201 or 1.5820368822243981e+217 < x

    1. Started with
      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
      13.9
    2. Using strategy rm
      13.9
    3. Applied add-exp-log to get
      \[{\color{red}{\left(x + 1\right)}}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)} \leadsto {\color{blue}{\left(e^{\log \left(x + 1\right)}\right)}}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
      14.0
    4. Applied pow-exp to get
      \[\color{red}{{\left(e^{\log \left(x + 1\right)}\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{1}{n}\right)} \leadsto \color{blue}{e^{\log \left(x + 1\right) \cdot \frac{1}{n}}} - {x}^{\left(\frac{1}{n}\right)}\]
      14.0
    5. Applied simplify to get
      \[e^{\color{red}{\log \left(x + 1\right) \cdot \frac{1}{n}}} - {x}^{\left(\frac{1}{n}\right)} \leadsto e^{\color{blue}{\frac{\log_* (1 + x)}{n}}} - {x}^{\left(\frac{1}{n}\right)}\]
      14.0
  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))))