\[{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}\]
Test:
NMSE problem 3.3.4
Bits:
128 bits
Bits error versus x
Time: 16.0 s
Input Error: 13.1
Output Error: 13.0
Log: ⚲
Profile: 🕒
\(\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} - \sqrt{\sqrt[3]{x}}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} + \sqrt{{x}^{\left(\frac{1}{3}\right)}}\right)\)
  1. Started with
    \[{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}\]
    13.1
  2. Using strategy rm
    13.1
  3. Applied add-sqr-sqrt to get
    \[{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - \color{red}{{x}^{\left(\frac{1}{3}\right)}} \leadsto {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - \color{blue}{{\left(\sqrt{{x}^{\left(\frac{1}{3}\right)}}\right)}^2}\]
    13.3
  4. Applied add-sqr-sqrt to get
    \[\color{red}{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} - {\left(\sqrt{{x}^{\left(\frac{1}{3}\right)}}\right)}^2 \leadsto \color{blue}{{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}}\right)}^2} - {\left(\sqrt{{x}^{\left(\frac{1}{3}\right)}}\right)}^2\]
    13.3
  5. Applied difference-of-squares to get
    \[\color{red}{{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}}\right)}^2 - {\left(\sqrt{{x}^{\left(\frac{1}{3}\right)}}\right)}^2} \leadsto \color{blue}{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} + \sqrt{{x}^{\left(\frac{1}{3}\right)}}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} - \sqrt{{x}^{\left(\frac{1}{3}\right)}}\right)}\]
    13.3
  6. Applied taylor to get
    \[\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} + \sqrt{{x}^{\left(\frac{1}{3}\right)}}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} - \sqrt{{x}^{\left(\frac{1}{3}\right)}}\right) \leadsto \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} + \sqrt{{x}^{\left(\frac{1}{3}\right)}}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} - \sqrt{{x}^{\frac{1}{3}}}\right)\]
    13.3
  7. Taylor expanded around 0 to get
    \[\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} + \sqrt{{x}^{\left(\frac{1}{3}\right)}}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} - \sqrt{\color{red}{{x}^{\frac{1}{3}}}}\right) \leadsto \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} + \sqrt{{x}^{\left(\frac{1}{3}\right)}}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} - \sqrt{\color{blue}{{x}^{\frac{1}{3}}}}\right)\]
    13.3
  8. Applied simplify to get
    \[\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} + \sqrt{{x}^{\left(\frac{1}{3}\right)}}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} - \sqrt{{x}^{\frac{1}{3}}}\right) \leadsto \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} - \sqrt{\sqrt[3]{x}}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} + \sqrt{{x}^{\left(\frac{1}{3}\right)}}\right)\]
    13.0
  9. Applied final simplification

Original test:


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