\[{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}\]
Test:
NMSE problem 3.3.4
Bits:
128 bits
Bits error versus x
Time: 3.8 s
Input Error: 21.4
Output Error: 12.8
Log: ⚲
Profile: 🕒
\(\sqrt[3]{1 + x} - \sqrt[3]{x}\)
  1. Started with
    \[{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}\]
    21.4
  2. Using strategy rm
    21.4
  3. Applied expm1-log1p-u to get
    \[\color{red}{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}} \leadsto \color{blue}{(e^{\log_* (1 + \left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}\right))} - 1)^*}\]
    21.4
  4. Applied taylor to get
    \[(e^{\log_* (1 + \left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}\right))} - 1)^* \leadsto (e^{\log_* (1 + \left({\left(1 + x\right)}^{\frac{1}{3}} - {x}^{\frac{1}{3}}\right))} - 1)^*\]
    21.4
  5. Taylor expanded around 0 to get
    \[\color{red}{(e^{\log_* (1 + \left({\left(1 + x\right)}^{\frac{1}{3}} - {x}^{\frac{1}{3}}\right))} - 1)^*} \leadsto \color{blue}{(e^{\log_* (1 + \left({\left(1 + x\right)}^{\frac{1}{3}} - {x}^{\frac{1}{3}}\right))} - 1)^*}\]
    21.4
  6. Applied simplify to get
    \[\color{red}{(e^{\log_* (1 + \left({\left(1 + x\right)}^{\frac{1}{3}} - {x}^{\frac{1}{3}}\right))} - 1)^*} \leadsto \color{blue}{\sqrt[3]{1 + x} - \sqrt[3]{x}}\]
    12.8
  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))))