\[{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}\]
Test:
NMSE problem 3.3.4
Bits:
128 bits
Bits error versus x
Time: 6.5 s
Input Error: 18.9
Output Error: 8.7
Log: ⚲
Profile: 🕒
\({\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}}\right)}^2 - \sqrt[3]{x}\)
  1. Started with
    \[{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}\]
    18.9
  2. Applied taylor to get
    \[{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)} \leadsto {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\frac{1}{3}}\]
    18.9
  3. Taylor expanded around 0 to get
    \[{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - \color{red}{{x}^{\frac{1}{3}}} \leadsto {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - \color{blue}{{x}^{\frac{1}{3}}}\]
    18.9
  4. Applied simplify to get
    \[\color{red}{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\frac{1}{3}}} \leadsto \color{blue}{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - \sqrt[3]{x}}\]
    8.7
  5. Using strategy rm
    8.7
  6. Applied add-sqr-sqrt to get
    \[\color{red}{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} - \sqrt[3]{x} \leadsto \color{blue}{{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}}\right)}^2} - \sqrt[3]{x}\]
    8.7
  7. Removed slow pow expressions

Original test:


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