\[\frac{x - \sin x}{x - \tan x}\]
Test:
NMSE problem 3.4.5
Bits:
128 bits
Bits error versus x
Time: 11.5 s
Input Error: 62.7
Output Error: 8.6
Log: ⚲
Profile: 🕒
\(\frac{{\left({x}^2 \cdot \frac{9}{40}\right)}^3 - {\left({x}^{4} \cdot \frac{27}{2800} + \frac{1}{2}\right)}^3}{{\left(\frac{9}{40} \cdot {x}^2\right)}^2 + \left({\left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)}^2 + \left(\frac{9}{40} \cdot {x}^2\right) \cdot \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)\right)}\)
  1. Started with
    \[\frac{x - \sin x}{x - \tan x}\]
    62.7
  2. Applied taylor to get
    \[\frac{x - \sin x}{x - \tan x} \leadsto \frac{9}{40} \cdot {x}^2 - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)\]
    0.1
  3. Taylor expanded around 0 to get
    \[\color{red}{\frac{9}{40} \cdot {x}^2 - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)} \leadsto \color{blue}{\frac{9}{40} \cdot {x}^2 - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)}\]
    0.1
  4. Using strategy rm
    0.1
  5. Applied flip3-- to get
    \[\color{red}{\frac{9}{40} \cdot {x}^2 - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)} \leadsto \color{blue}{\frac{{\left(\frac{9}{40} \cdot {x}^2\right)}^{3} - {\left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)}^{3}}{{\left(\frac{9}{40} \cdot {x}^2\right)}^2 + \left({\left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)}^2 + \left(\frac{9}{40} \cdot {x}^2\right) \cdot \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)\right)}}\]
    8.6
  6. Applied simplify to get
    \[\frac{\color{red}{{\left(\frac{9}{40} \cdot {x}^2\right)}^{3} - {\left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)}^{3}}}{{\left(\frac{9}{40} \cdot {x}^2\right)}^2 + \left({\left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)}^2 + \left(\frac{9}{40} \cdot {x}^2\right) \cdot \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)\right)} \leadsto \frac{\color{blue}{{\left(\left(x \cdot x\right) \cdot \frac{9}{40}\right)}^3 - {\left({x}^{4} \cdot \frac{27}{2800} + \frac{1}{2}\right)}^3}}{{\left(\frac{9}{40} \cdot {x}^2\right)}^2 + \left({\left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)}^2 + \left(\frac{9}{40} \cdot {x}^2\right) \cdot \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)\right)}\]
    8.6
  7. Applied simplify to get
    \[\frac{\color{red}{{\left(\left(x \cdot x\right) \cdot \frac{9}{40}\right)}^3} - {\left({x}^{4} \cdot \frac{27}{2800} + \frac{1}{2}\right)}^3}{{\left(\frac{9}{40} \cdot {x}^2\right)}^2 + \left({\left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)}^2 + \left(\frac{9}{40} \cdot {x}^2\right) \cdot \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)\right)} \leadsto \frac{\color{blue}{{\left({x}^2 \cdot \frac{9}{40}\right)}^3} - {\left({x}^{4} \cdot \frac{27}{2800} + \frac{1}{2}\right)}^3}{{\left(\frac{9}{40} \cdot {x}^2\right)}^2 + \left({\left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)}^2 + \left(\frac{9}{40} \cdot {x}^2\right) \cdot \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)\right)}\]
    8.6

Original test:


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