\[{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}\]
Test:
NMSE problem 3.3.4
Bits:
128 bits
Bits error versus x
Time: 16.7 s
Input Error: 40.7
Output Error: 1.7
Log: ⚲
Profile: 🕒
\(\frac{1 - \left(\frac{1}{x} + \frac{-1}{x}\right)}{(\left({\left(1 + x\right)}^{\left(\frac{1}{3}\right)}\right) * \left({\left(1 + x\right)}^{\left(\frac{1}{3}\right)} + \sqrt[3]{x}\right) + \left({\left(\sqrt[3]{x}\right)}^2\right))_*}\)
  1. Started with
    \[{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}\]
    40.7
  2. Using strategy rm
    40.7
  3. Applied flip3-- to get
    \[\color{red}{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}} \leadsto \color{blue}{\frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^{3} - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^{3}}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right)}}\]
    40.7
  4. Applied simplify to get
    \[\frac{\color{red}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^{3} - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^{3}}}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right)} \leadsto \frac{\color{blue}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^3 - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^3}}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right)}\]
    40.6
  5. Applied taylor to get
    \[\frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^3 - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^3}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right)} \leadsto \frac{{\left({\left(1 - \frac{1}{x}\right)}^{\frac{1}{3}}\right)}^3 - {\left({\left(\frac{-1}{x}\right)}^{\frac{1}{3}}\right)}^3}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right)}\]
    62.1
  6. Taylor expanded around -inf to get
    \[\frac{\color{red}{{\left({\left(1 - \frac{1}{x}\right)}^{\frac{1}{3}}\right)}^3 - {\left({\left(\frac{-1}{x}\right)}^{\frac{1}{3}}\right)}^3}}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right)} \leadsto \frac{\color{blue}{{\left({\left(1 - \frac{1}{x}\right)}^{\frac{1}{3}}\right)}^3 - {\left({\left(\frac{-1}{x}\right)}^{\frac{1}{3}}\right)}^3}}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right)}\]
    62.1
  7. Applied simplify to get
    \[\color{red}{\frac{{\left({\left(1 - \frac{1}{x}\right)}^{\frac{1}{3}}\right)}^3 - {\left({\left(\frac{-1}{x}\right)}^{\frac{1}{3}}\right)}^3}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right)}} \leadsto \color{blue}{\frac{1 - \left(\frac{1}{x} + \frac{-1}{x}\right)}{(\left({\left(1 + x\right)}^{\left(\frac{1}{3}\right)}\right) * \left({x}^{\left(\frac{1}{3}\right)} + {\left(1 + x\right)}^{\left(\frac{1}{3}\right)}\right) + \left({x}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right))_*}}\]
    22.9
  8. Applied taylor to get
    \[\frac{1 - \left(\frac{1}{x} + \frac{-1}{x}\right)}{(\left({\left(1 + x\right)}^{\left(\frac{1}{3}\right)}\right) * \left({x}^{\left(\frac{1}{3}\right)} + {\left(1 + x\right)}^{\left(\frac{1}{3}\right)}\right) + \left({x}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right))_*} \leadsto \frac{1 - \left(\frac{1}{x} + \frac{-1}{x}\right)}{(\left({\left(1 + x\right)}^{\left(\frac{1}{3}\right)}\right) * \left({x}^{\frac{1}{3}} + {\left(1 + x\right)}^{\left(\frac{1}{3}\right)}\right) + \left({x}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right))_*}\]
    22.9
  9. Taylor expanded around 0 to get
    \[\frac{1 - \left(\frac{1}{x} + \frac{-1}{x}\right)}{(\left({\left(1 + x\right)}^{\left(\frac{1}{3}\right)}\right) * \left(\color{red}{{x}^{\frac{1}{3}}} + {\left(1 + x\right)}^{\left(\frac{1}{3}\right)}\right) + \left({x}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right))_*} \leadsto \frac{1 - \left(\frac{1}{x} + \frac{-1}{x}\right)}{(\left({\left(1 + x\right)}^{\left(\frac{1}{3}\right)}\right) * \left(\color{blue}{{x}^{\frac{1}{3}}} + {\left(1 + x\right)}^{\left(\frac{1}{3}\right)}\right) + \left({x}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right))_*}\]
    22.9
  10. Applied simplify to get
    \[\color{red}{\frac{1 - \left(\frac{1}{x} + \frac{-1}{x}\right)}{(\left({\left(1 + x\right)}^{\left(\frac{1}{3}\right)}\right) * \left({x}^{\frac{1}{3}} + {\left(1 + x\right)}^{\left(\frac{1}{3}\right)}\right) + \left({x}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right))_*}} \leadsto \color{blue}{\frac{1 - \left(\frac{-1}{x} + \frac{1}{x}\right)}{(\left({\left(1 + x\right)}^{\left(\frac{1}{3}\right)}\right) * \left(\sqrt[3]{x} + {\left(1 + x\right)}^{\left(\frac{1}{3}\right)}\right) + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2\right))_*}}\]
    22.8
  11. Applied taylor to get
    \[\frac{1 - \left(\frac{-1}{x} + \frac{1}{x}\right)}{(\left({\left(1 + x\right)}^{\left(\frac{1}{3}\right)}\right) * \left(\sqrt[3]{x} + {\left(1 + x\right)}^{\left(\frac{1}{3}\right)}\right) + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2\right))_*} \leadsto \frac{1 - \left(\frac{-1}{x} + \frac{1}{x}\right)}{(\left({\left(1 + x\right)}^{\left(\frac{1}{3}\right)}\right) * \left(\sqrt[3]{x} + {\left(1 + x\right)}^{\left(\frac{1}{3}\right)}\right) + \left({\left({x}^{\frac{1}{3}}\right)}^2\right))_*}\]
    22.8
  12. Taylor expanded around 0 to get
    \[\frac{1 - \left(\frac{-1}{x} + \frac{1}{x}\right)}{(\left({\left(1 + x\right)}^{\left(\frac{1}{3}\right)}\right) * \left(\sqrt[3]{x} + {\left(1 + x\right)}^{\left(\frac{1}{3}\right)}\right) + \left({\color{red}{\left({x}^{\frac{1}{3}}\right)}}^2\right))_*} \leadsto \frac{1 - \left(\frac{-1}{x} + \frac{1}{x}\right)}{(\left({\left(1 + x\right)}^{\left(\frac{1}{3}\right)}\right) * \left(\sqrt[3]{x} + {\left(1 + x\right)}^{\left(\frac{1}{3}\right)}\right) + \left({\color{blue}{\left({x}^{\frac{1}{3}}\right)}}^2\right))_*}\]
    22.8
  13. Applied simplify to get
    \[\frac{1 - \left(\frac{-1}{x} + \frac{1}{x}\right)}{(\left({\left(1 + x\right)}^{\left(\frac{1}{3}\right)}\right) * \left(\sqrt[3]{x} + {\left(1 + x\right)}^{\left(\frac{1}{3}\right)}\right) + \left({\left({x}^{\frac{1}{3}}\right)}^2\right))_*} \leadsto \frac{1 - \left(\frac{1}{x} + \frac{-1}{x}\right)}{(\left({\left(1 + x\right)}^{\left(\frac{1}{3}\right)}\right) * \left({\left(1 + x\right)}^{\left(\frac{1}{3}\right)} + \sqrt[3]{x}\right) + \left({\left(\sqrt[3]{x}\right)}^2\right))_*}\]
    1.7
  14. Applied final simplification
  15. Removed slow pow expressions

Original test:


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