\[{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}\]
Test:
NMSE problem 3.3.4
Bits:
128 bits
Bits error versus x
Time: 7.9 s
Input Error: 13.6
Output Error: 13.7
Log: ⚲
Profile: 🕒
\(\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}}\right)\)
  1. Started with
    \[{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}\]
    13.6
  2. Using strategy rm
    13.6
  3. Applied add-log-exp to get
    \[{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - \color{red}{{x}^{\left(\frac{1}{3}\right)}} \leadsto {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - \color{blue}{\log \left(e^{{x}^{\left(\frac{1}{3}\right)}}\right)}\]
    14.8
  4. Applied add-log-exp to get
    \[\color{red}{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} - \log \left(e^{{x}^{\left(\frac{1}{3}\right)}}\right) \leadsto \color{blue}{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}}\right)} - \log \left(e^{{x}^{\left(\frac{1}{3}\right)}}\right)\]
    14.4
  5. Applied diff-log to get
    \[\color{red}{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}}\right) - \log \left(e^{{x}^{\left(\frac{1}{3}\right)}}\right)} \leadsto \color{blue}{\log \left(\frac{e^{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}}}{e^{{x}^{\left(\frac{1}{3}\right)}}}\right)}\]
    14.5
  6. Applied simplify to get
    \[\log \color{red}{\left(\frac{e^{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}}}{e^{{x}^{\left(\frac{1}{3}\right)}}}\right)} \leadsto \log \color{blue}{\left(e^{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}}\right)}\]
    13.7

Original test:


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