\[{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}\]
Test:
NMSE problem 3.3.4
Bits:
128 bits
Bits error versus x
Time: 15.3 s
Input Error: 30.0
Output Error: 29.9
Log: ⚲
Profile: 🕒
\(\frac{{\left(e^{\frac{\log_* (1 + x)}{3}}\right)}^3 - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^3}{{\left(e^{\frac{\log_* (1 + x)}{3}}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + e^{\frac{\log_* (1 + x)}{3}} \cdot {x}^{\left(\frac{1}{3}\right)}\right)}\)
  1. Started with
    \[{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}\]
    30.0
  2. Using strategy rm
    30.0
  3. Applied add-exp-log to get
    \[{\color{red}{\left(x + 1\right)}}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)} \leadsto {\color{blue}{\left(e^{\log \left(x + 1\right)}\right)}}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}\]
    30.2
  4. Applied pow-exp to get
    \[\color{red}{{\left(e^{\log \left(x + 1\right)}\right)}^{\left(\frac{1}{3}\right)}} - {x}^{\left(\frac{1}{3}\right)} \leadsto \color{blue}{e^{\log \left(x + 1\right) \cdot \frac{1}{3}}} - {x}^{\left(\frac{1}{3}\right)}\]
    30.2
  5. Applied simplify to get
    \[e^{\color{red}{\log \left(x + 1\right) \cdot \frac{1}{3}}} - {x}^{\left(\frac{1}{3}\right)} \leadsto e^{\color{blue}{\frac{\log_* (1 + x)}{3}}} - {x}^{\left(\frac{1}{3}\right)}\]
    29.9
  6. Using strategy rm
    29.9
  7. Applied flip3-- to get
    \[\color{red}{e^{\frac{\log_* (1 + x)}{3}} - {x}^{\left(\frac{1}{3}\right)}} \leadsto \color{blue}{\frac{{\left(e^{\frac{\log_* (1 + x)}{3}}\right)}^{3} - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^{3}}{{\left(e^{\frac{\log_* (1 + x)}{3}}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + e^{\frac{\log_* (1 + x)}{3}} \cdot {x}^{\left(\frac{1}{3}\right)}\right)}}\]
    29.9
  8. Applied simplify to get
    \[\frac{\color{red}{{\left(e^{\frac{\log_* (1 + x)}{3}}\right)}^{3} - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^{3}}}{{\left(e^{\frac{\log_* (1 + x)}{3}}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + e^{\frac{\log_* (1 + x)}{3}} \cdot {x}^{\left(\frac{1}{3}\right)}\right)} \leadsto \frac{\color{blue}{{\left(e^{\frac{\log_* (1 + x)}{3}}\right)}^3 - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^3}}{{\left(e^{\frac{\log_* (1 + x)}{3}}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + e^{\frac{\log_* (1 + x)}{3}} \cdot {x}^{\left(\frac{1}{3}\right)}\right)}\]
    29.9
  9. Removed slow pow expressions

Original test:


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