\[{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}\]
Test:
NMSE problem 3.3.4
Bits:
128 bits
Bits error versus x
Time: 20.3 s
Input Error: 9.5
Output Error: 12.7
Log: ⚲
Profile: 🕒
\(\begin{cases} {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\frac{1}{6}} \cdot {x}^{\frac{1}{6}} & \text{when } x \le 2.902945f+08 \\ \left(\sqrt[3]{\frac{1}{x}} - \frac{1}{9} \cdot \sqrt[3]{\frac{1}{{x}^{7}}}\right) - \left({x}^{\frac{-1}{3}} - \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{4}}}\right) & \text{otherwise} \end{cases}\)

    if x < 2.902945f+08

    1. Started with
      \[{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}\]
      2.6
    2. Using strategy rm
      2.6
    3. Applied add-sqr-sqrt to get
      \[{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - \color{red}{{x}^{\left(\frac{1}{3}\right)}} \leadsto {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - \color{blue}{{\left(\sqrt{{x}^{\left(\frac{1}{3}\right)}}\right)}^2}\]
      2.7
    4. Applied taylor to get
      \[{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {\left(\sqrt{{x}^{\left(\frac{1}{3}\right)}}\right)}^2 \leadsto {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {\left({x}^{\frac{1}{6}}\right)}^2\]
      2.7
    5. Taylor expanded around 0 to get
      \[{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {\color{red}{\left({x}^{\frac{1}{6}}\right)}}^2 \leadsto {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {\color{blue}{\left({x}^{\frac{1}{6}}\right)}}^2\]
      2.7
    6. Applied simplify to get
      \[{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {\left({x}^{\frac{1}{6}}\right)}^2 \leadsto {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\frac{1}{6}} \cdot {x}^{\frac{1}{6}}\]
      2.7
    7. Applied final simplification

    if 2.902945f+08 < x

    1. Started with
      \[{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}\]
      20.3
    2. Using strategy rm
      20.3
    3. Applied add-sqr-sqrt to get
      \[\color{red}{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} - {x}^{\left(\frac{1}{3}\right)} \leadsto \color{blue}{{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}}\right)}^2} - {x}^{\left(\frac{1}{3}\right)}\]
      29.9
    4. Applied taylor to get
      \[{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}}\right)}^2 - {x}^{\left(\frac{1}{3}\right)} \leadsto \left(\frac{1}{3} \cdot {\left(\frac{1}{{x}^{4}}\right)}^{\frac{1}{3}} + {\left(\frac{1}{x}\right)}^{\frac{1}{3}}\right) - \left(\frac{1}{9} \cdot {\left(\frac{1}{{x}^{7}}\right)}^{\frac{1}{3}} + {x}^{\frac{-1}{3}}\right)\]
      20.2
    5. Taylor expanded around inf to get
      \[\color{red}{\left(\frac{1}{3} \cdot {\left(\frac{1}{{x}^{4}}\right)}^{\frac{1}{3}} + {\left(\frac{1}{x}\right)}^{\frac{1}{3}}\right) - \left(\frac{1}{9} \cdot {\left(\frac{1}{{x}^{7}}\right)}^{\frac{1}{3}} + {x}^{\frac{-1}{3}}\right)} \leadsto \color{blue}{\left(\frac{1}{3} \cdot {\left(\frac{1}{{x}^{4}}\right)}^{\frac{1}{3}} + {\left(\frac{1}{x}\right)}^{\frac{1}{3}}\right) - \left(\frac{1}{9} \cdot {\left(\frac{1}{{x}^{7}}\right)}^{\frac{1}{3}} + {x}^{\frac{-1}{3}}\right)}\]
      20.2
    6. Applied simplify to get
      \[\left(\frac{1}{3} \cdot {\left(\frac{1}{{x}^{4}}\right)}^{\frac{1}{3}} + {\left(\frac{1}{x}\right)}^{\frac{1}{3}}\right) - \left(\frac{1}{9} \cdot {\left(\frac{1}{{x}^{7}}\right)}^{\frac{1}{3}} + {x}^{\frac{-1}{3}}\right) \leadsto \left(\sqrt[3]{\frac{1}{x}} - \frac{1}{9} \cdot \sqrt[3]{\frac{1}{{x}^{7}}}\right) - \left({x}^{\frac{-1}{3}} - \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{4}}}\right)\]
      28.5
    7. Applied final simplification
  1. Removed slow pow expressions

Original test:


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