\[\frac{x - \sin x}{x - \tan x}\]
Test:
NMSE problem 3.4.5
Bits:
128 bits
Bits error versus x
Time: 6.7 s
Input Error: 29.9
Output Error: 0.3
Log: ⚲
Profile: 🕒
\({x}^2 \cdot \frac{9}{40} - (\frac{27}{2800} * \left({x}^{4}\right) + \frac{1}{2})_*\)
  1. Started with
    \[\frac{x - \sin x}{x - \tan x}\]
    29.9
  2. Applied taylor to get
    \[\frac{x - \sin x}{x - \tan x} \leadsto \frac{9}{40} \cdot {x}^2 - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)\]
    0.3
  3. Taylor expanded around 0 to get
    \[\color{red}{\frac{9}{40} \cdot {x}^2 - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)} \leadsto \color{blue}{\frac{9}{40} \cdot {x}^2 - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)}\]
    0.3
  4. Using strategy rm
    0.3
  5. Applied add-cube-cbrt to get
    \[\color{red}{\frac{9}{40} \cdot {x}^2 - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)} \leadsto \color{blue}{{\left(\sqrt[3]{\frac{9}{40} \cdot {x}^2 - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)}\right)}^3}\]
    1.2
  6. Applied taylor to get
    \[{\left(\sqrt[3]{\frac{9}{40} \cdot {x}^2 - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)}\right)}^3 \leadsto {\left(\sqrt[3]{\frac{9}{40} \cdot {x}^2 - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)}\right)}^3\]
    1.2
  7. Taylor expanded around 0 to get
    \[{\left(\sqrt[3]{\color{red}{\frac{9}{40} \cdot {x}^2 - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)}}\right)}^3 \leadsto {\left(\sqrt[3]{\color{blue}{\frac{9}{40} \cdot {x}^2 - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)}}\right)}^3\]
    1.2
  8. Applied simplify to get
    \[\color{red}{{\left(\sqrt[3]{\frac{9}{40} \cdot {x}^2 - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)}\right)}^3} \leadsto \color{blue}{\left(x \cdot x\right) \cdot \frac{9}{40} - (\frac{27}{2800} * \left({x}^{4}\right) + \frac{1}{2})_*}\]
    0.3
  9. Applied simplify to get
    \[\color{red}{\left(x \cdot x\right) \cdot \frac{9}{40}} - (\frac{27}{2800} * \left({x}^{4}\right) + \frac{1}{2})_* \leadsto \color{blue}{{x}^2 \cdot \frac{9}{40}} - (\frac{27}{2800} * \left({x}^{4}\right) + \frac{1}{2})_*\]
    0.3

Original test:


(lambda ((x default))
  #:name "NMSE problem 3.4.5"
  (/ (- x (sin x)) (- x (tan x))))