\[{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}\]
Test:
NMSE problem 3.3.4
Bits:
128 bits
Bits error versus x
Time: 20.7 s
Input Error: 30.4
Output Error: 30.4
Log: ⚲
Profile: 🕒
\(\frac{{\left(\sqrt[3]{{\left(e^{\frac{\log_* (1 + x)}{3}}\right)}^3}\right)}^2 - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^2}{{x}^{\left(\frac{1}{3}\right)} + e^{\frac{\log_* (1 + x)}{3}}}\)
  1. Started with
    \[{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}\]
    30.4
  2. Using strategy rm
    30.4
  3. Applied add-exp-log to get
    \[{\color{red}{\left(x + 1\right)}}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)} \leadsto {\color{blue}{\left(e^{\log \left(x + 1\right)}\right)}}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}\]
    30.7
  4. Applied pow-exp to get
    \[\color{red}{{\left(e^{\log \left(x + 1\right)}\right)}^{\left(\frac{1}{3}\right)}} - {x}^{\left(\frac{1}{3}\right)} \leadsto \color{blue}{e^{\log \left(x + 1\right) \cdot \frac{1}{3}}} - {x}^{\left(\frac{1}{3}\right)}\]
    30.7
  5. Applied simplify to get
    \[e^{\color{red}{\log \left(x + 1\right) \cdot \frac{1}{3}}} - {x}^{\left(\frac{1}{3}\right)} \leadsto e^{\color{blue}{\frac{\log_* (1 + x)}{3}}} - {x}^{\left(\frac{1}{3}\right)}\]
    30.4
  6. Using strategy rm
    30.4
  7. Applied add-cbrt-cube to get
    \[\color{red}{e^{\frac{\log_* (1 + x)}{3}}} - {x}^{\left(\frac{1}{3}\right)} \leadsto \color{blue}{\sqrt[3]{{\left(e^{\frac{\log_* (1 + x)}{3}}\right)}^3}} - {x}^{\left(\frac{1}{3}\right)}\]
    30.4
  8. Using strategy rm
    30.4
  9. Applied flip-- to get
    \[\color{red}{\sqrt[3]{{\left(e^{\frac{\log_* (1 + x)}{3}}\right)}^3} - {x}^{\left(\frac{1}{3}\right)}} \leadsto \color{blue}{\frac{{\left(\sqrt[3]{{\left(e^{\frac{\log_* (1 + x)}{3}}\right)}^3}\right)}^2 - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^2}{\sqrt[3]{{\left(e^{\frac{\log_* (1 + x)}{3}}\right)}^3} + {x}^{\left(\frac{1}{3}\right)}}}\]
    30.4
  10. Applied simplify to get
    \[\frac{{\left(\sqrt[3]{{\left(e^{\frac{\log_* (1 + x)}{3}}\right)}^3}\right)}^2 - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^2}{\color{red}{\sqrt[3]{{\left(e^{\frac{\log_* (1 + x)}{3}}\right)}^3} + {x}^{\left(\frac{1}{3}\right)}}} \leadsto \frac{{\left(\sqrt[3]{{\left(e^{\frac{\log_* (1 + x)}{3}}\right)}^3}\right)}^2 - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^2}{\color{blue}{{x}^{\left(\frac{1}{3}\right)} + e^{\frac{\log_* (1 + x)}{3}}}}\]
    30.4

Original test:


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