\[{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}\]
Test:
NMSE problem 3.3.4
Bits:
128 bits
Bits error versus x
Time: 10.5 s
Input Error: 22.0
Output Error: 13.2
Log: ⚲
Profile: 🕒
\(\sqrt[3]{1 + x} - \sqrt[3]{x}\)
  1. Started with
    \[{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}\]
    22.0
  2. Using strategy rm
    22.0
  3. Applied add-cube-cbrt to get
    \[\color{red}{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} - {x}^{\left(\frac{1}{3}\right)} \leadsto \color{blue}{{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}}\right)}^3} - {x}^{\left(\frac{1}{3}\right)}\]
    22.1
  4. Applied taylor to get
    \[{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}}\right)}^3 - {x}^{\left(\frac{1}{3}\right)} \leadsto {\left(1 + x\right)}^{\frac{1}{3}} - {x}^{\frac{1}{3}}\]
    22.0
  5. Taylor expanded around 0 to get
    \[\color{red}{{\left(1 + x\right)}^{\frac{1}{3}} - {x}^{\frac{1}{3}}} \leadsto \color{blue}{{\left(1 + x\right)}^{\frac{1}{3}} - {x}^{\frac{1}{3}}}\]
    22.0
  6. Applied simplify to get
    \[{\left(1 + x\right)}^{\frac{1}{3}} - {x}^{\frac{1}{3}} \leadsto \sqrt[3]{1 + x} - \sqrt[3]{x}\]
    13.2
  7. Applied final simplification

Original test:


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