\[{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}\]
Test:
NMSE problem 3.3.4
Bits:
128 bits
Bits error versus x
Time: 6.8 s
Input Error: 45.3
Output Error: 29.5
Log: ⚲
Profile: 🕒
\(\sqrt[3]{1 + x} - \sqrt[3]{x}\)
  1. Started with
    \[{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}\]
    45.3
  2. Using strategy rm
    45.3
  3. Applied add-cbrt-cube to get
    \[\color{red}{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} - {x}^{\left(\frac{1}{3}\right)} \leadsto \color{blue}{\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^3}} - {x}^{\left(\frac{1}{3}\right)}\]
    45.3
  4. Applied taylor to get
    \[\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^3} - {x}^{\left(\frac{1}{3}\right)} \leadsto {\left(1 + x\right)}^{\frac{1}{3}} - {x}^{\frac{1}{3}}\]
    45.3
  5. Taylor expanded around 0 to get
    \[\color{red}{{\left(1 + x\right)}^{\frac{1}{3}} - {x}^{\frac{1}{3}}} \leadsto \color{blue}{{\left(1 + x\right)}^{\frac{1}{3}} - {x}^{\frac{1}{3}}}\]
    45.3
  6. Applied simplify to get
    \[\color{red}{{\left(1 + x\right)}^{\frac{1}{3}} - {x}^{\frac{1}{3}}} \leadsto \color{blue}{\sqrt[3]{1 + x} - \sqrt[3]{x}}\]
    29.5
  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))))