\[{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}\]
Test:
NMSE problem 3.3.4
Bits:
128 bits
Bits error versus x
Time: 10.5 s
Input Error: 21.4
Output Error: 12.8
Log: ⚲
Profile: 🕒
\(\sqrt[3]{1 + x} - \sqrt[3]{x}\)
  1. Started with
    \[{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}\]
    21.4
  2. Using strategy rm
    21.4
  3. Applied add-log-exp to get
    \[{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - \color{red}{{x}^{\left(\frac{1}{3}\right)}} \leadsto {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - \color{blue}{\log \left(e^{{x}^{\left(\frac{1}{3}\right)}}\right)}\]
    21.9
  4. Applied add-log-exp to get
    \[\color{red}{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} - \log \left(e^{{x}^{\left(\frac{1}{3}\right)}}\right) \leadsto \color{blue}{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}}\right)} - \log \left(e^{{x}^{\left(\frac{1}{3}\right)}}\right)\]
    21.8
  5. Applied diff-log to get
    \[\color{red}{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}}\right) - \log \left(e^{{x}^{\left(\frac{1}{3}\right)}}\right)} \leadsto \color{blue}{\log \left(\frac{e^{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}}}{e^{{x}^{\left(\frac{1}{3}\right)}}}\right)}\]
    21.8
  6. Applied simplify to get
    \[\log \color{red}{\left(\frac{e^{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}}}{e^{{x}^{\left(\frac{1}{3}\right)}}}\right)} \leadsto \log \color{blue}{\left(e^{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}}\right)}\]
    21.4
  7. Using strategy rm
    21.4
  8. Applied add-cube-cbrt to get
    \[\log \color{red}{\left(e^{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}}\right)} \leadsto \log \color{blue}{\left({\left(\sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}}}\right)}^3\right)}\]
    21.7
  9. Applied taylor to get
    \[\log \left({\left(\sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}}}\right)}^3\right) \leadsto \log \left(e^{{\left(1 + x\right)}^{\frac{1}{3}} - {x}^{\frac{1}{3}}}\right)\]
    21.4
  10. Taylor expanded around 0 to get
    \[\log \color{red}{\left(e^{{\left(1 + x\right)}^{\frac{1}{3}} - {x}^{\frac{1}{3}}}\right)} \leadsto \log \color{blue}{\left(e^{{\left(1 + x\right)}^{\frac{1}{3}} - {x}^{\frac{1}{3}}}\right)}\]
    21.4
  11. Applied simplify to get
    \[\log \left(e^{{\left(1 + x\right)}^{\frac{1}{3}} - {x}^{\frac{1}{3}}}\right) \leadsto \sqrt[3]{1 + x} - \sqrt[3]{x}\]
    12.8
  12. Applied final simplification

Original test:


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