\[{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}\]
Test:
NMSE problem 3.3.4
Bits:
128 bits
Bits error versus x
Time: 11.5 s
Input Error: 13.1
Output Error: 13.1
Log: ⚲
Profile: 🕒
\(e^{\log \left({\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}\right)}^3\right) \cdot \frac{1}{3}}\)
  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-cbrt-cube to get
    \[\color{red}{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}} \leadsto \color{blue}{\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}\right)}^3}}\]
    13.1
  4. Using strategy rm
    13.1
  5. Applied pow1/3 to get
    \[\color{red}{\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}\right)}^3}} \leadsto \color{blue}{{\left({\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}\right)}^3\right)}^{\frac{1}{3}}}\]
    13.1
  6. Using strategy rm
    13.1
  7. Applied add-exp-log to get
    \[{\color{red}{\left({\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}\right)}^3\right)}}^{\frac{1}{3}} \leadsto {\color{blue}{\left(e^{\log \left({\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}\right)}^3\right)}\right)}}^{\frac{1}{3}}\]
    13.1
  8. Applied pow-exp to get
    \[\color{red}{{\left(e^{\log \left({\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}\right)}^3\right)}\right)}^{\frac{1}{3}}} \leadsto \color{blue}{e^{\log \left({\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}\right)}^3\right) \cdot \frac{1}{3}}}\]
    13.1

Original test:


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