\[{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}\]
Test:
NMSE problem 3.3.4
Bits:
128 bits
Bits error versus x
Time: 4.1 s
Input Error: 21.7
Output Error: 12.8
Log: ⚲
Profile: 🕒
\(\sqrt[3]{1 + x} - \sqrt[3]{x}\)
  1. Started with
    \[{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}\]
    21.7
  2. Using strategy rm
    21.7
  3. Applied add-exp-log to get
    \[\color{red}{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}} \leadsto \color{blue}{e^{\log \left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}\right)}}\]
    21.7
  4. Using strategy rm
    21.7
  5. Applied add-cube-cbrt to get
    \[e^{\color{red}{\log \left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}\right)}} \leadsto e^{\color{blue}{{\left(\sqrt[3]{\log \left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}\right)}\right)}^3}}\]
    21.7
  6. Applied taylor to get
    \[e^{{\left(\sqrt[3]{\log \left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}\right)}\right)}^3} \leadsto e^{\log \left({\left(1 + x\right)}^{\frac{1}{3}} - {x}^{\frac{1}{3}}\right)}\]
    21.7
  7. Taylor expanded around 0 to get
    \[e^{\color{red}{\log \left({\left(1 + x\right)}^{\frac{1}{3}} - {x}^{\frac{1}{3}}\right)}} \leadsto e^{\color{blue}{\log \left({\left(1 + x\right)}^{\frac{1}{3}} - {x}^{\frac{1}{3}}\right)}}\]
    21.7
  8. Applied simplify to get
    \[\color{red}{e^{\log \left({\left(1 + x\right)}^{\frac{1}{3}} - {x}^{\frac{1}{3}}\right)}} \leadsto \color{blue}{\sqrt[3]{1 + x} - \sqrt[3]{x}}\]
    12.8

Original test:


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