\[{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}\]
Test:
NMSE problem 3.3.4
Bits:
128 bits
Bits error versus x
Time: 15.0 s
Input Error: 13.2
Output Error: 13.1
Log: ⚲
Profile: 🕒
\(\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} + \sqrt{e^{\frac{\log x}{3}}}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} - \sqrt{\sqrt[3]{x}}\right)\)
  1. Started with
    \[{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}\]
    13.2
  2. Using strategy rm
    13.2
  3. Applied pow-to-exp to get
    \[{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - \color{red}{{x}^{\left(\frac{1}{3}\right)}} \leadsto {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - \color{blue}{e^{\log x \cdot \frac{1}{3}}}\]
    13.5
  4. Applied simplify to get
    \[{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - e^{\color{red}{\log x \cdot \frac{1}{3}}} \leadsto {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - e^{\color{blue}{\frac{\log x}{3}}}\]
    13.2
  5. Using strategy rm
    13.2
  6. Applied add-sqr-sqrt to get
    \[{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - \color{red}{e^{\frac{\log x}{3}}} \leadsto {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - \color{blue}{{\left(\sqrt{e^{\frac{\log x}{3}}}\right)}^2}\]
    13.3
  7. Applied add-sqr-sqrt to get
    \[\color{red}{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} - {\left(\sqrt{e^{\frac{\log x}{3}}}\right)}^2 \leadsto \color{blue}{{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}}\right)}^2} - {\left(\sqrt{e^{\frac{\log x}{3}}}\right)}^2\]
    13.3
  8. Applied difference-of-squares to get
    \[\color{red}{{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}}\right)}^2 - {\left(\sqrt{e^{\frac{\log x}{3}}}\right)}^2} \leadsto \color{blue}{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} + \sqrt{e^{\frac{\log x}{3}}}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} - \sqrt{e^{\frac{\log x}{3}}}\right)}\]
    13.3
  9. Applied taylor to get
    \[\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} + \sqrt{e^{\frac{\log x}{3}}}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} - \sqrt{e^{\frac{\log x}{3}}}\right) \leadsto \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} + \sqrt{e^{\frac{\log x}{3}}}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} - \sqrt{{x}^{\frac{1}{3}}}\right)\]
    13.4
  10. Taylor expanded around 0 to get
    \[\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} + \sqrt{e^{\frac{\log x}{3}}}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} - \sqrt{\color{red}{{x}^{\frac{1}{3}}}}\right) \leadsto \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} + \sqrt{e^{\frac{\log x}{3}}}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} - \sqrt{\color{blue}{{x}^{\frac{1}{3}}}}\right)\]
    13.4
  11. Applied simplify to get
    \[\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} + \sqrt{e^{\frac{\log x}{3}}}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} - \sqrt{{x}^{\frac{1}{3}}}\right) \leadsto \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} + \sqrt{e^{\frac{\log x}{3}}}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} - \sqrt{\sqrt[3]{x}}\right)\]
    13.1
  12. Applied final simplification

Original test:


(lambda ((x default))
  #:name "NMSE problem 3.3.4"
  (- (pow (+ x 1) (/ 1 3)) (pow x (/ 1 3))))