\[\frac{x - \sin x}{x - \tan x}\]
Test:
NMSE problem 3.4.5
Bits:
128 bits
Bits error versus x
Time: 22.4 s
Input Error: 30.9
Output Error: 0.0
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x} & \text{when } x \le -0.025556113738953053 \\ \left({x}^2 \cdot \frac{9}{40} - \frac{27}{2800} \cdot {x}^{4}\right) - \frac{1}{2} & \text{when } x \le 17.12856537901564 \\ \frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x} & \text{otherwise} \end{cases}\)

    if x < -0.025556113738953053 or 17.12856537901564 < x

    1. Started with
      \[\frac{x - \sin x}{x - \tan x}\]
      0.0
    2. Using strategy rm
      0.0
    3. Applied div-sub to get
      \[\color{red}{\frac{x - \sin x}{x - \tan x}} \leadsto \color{blue}{\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}}\]
      0.0

    if -0.025556113738953053 < x < 17.12856537901564

    1. Started with
      \[\frac{x - \sin x}{x - \tan x}\]
      62.7
    2. Applied taylor to get
      \[\frac{x - \sin x}{x - \tan x} \leadsto \frac{9}{40} \cdot {x}^2 - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)\]
      0.0
    3. Taylor expanded around 0 to get
      \[\color{red}{\frac{9}{40} \cdot {x}^2 - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)} \leadsto \color{blue}{\frac{9}{40} \cdot {x}^2 - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)}\]
      0.0
    4. Applied taylor to get
      \[\frac{9}{40} \cdot {x}^2 - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right) \leadsto \frac{9}{40} \cdot {x}^2 - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)\]
      0.0
    5. Taylor expanded around 0 to get
      \[\color{red}{\frac{9}{40} \cdot {x}^2 - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)} \leadsto \color{blue}{\frac{9}{40} \cdot {x}^2 - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)}\]
      0.0
    6. Applied simplify to get
      \[\frac{9}{40} \cdot {x}^2 - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right) \leadsto \left(\left(x \cdot x\right) \cdot \frac{9}{40} - \frac{27}{2800} \cdot {x}^{4}\right) - \frac{1}{2}\]
      0.0
    7. Applied final simplification
    8. Applied simplify to get
      \[\color{red}{\left(\left(x \cdot x\right) \cdot \frac{9}{40} - \frac{27}{2800} \cdot {x}^{4}\right) - \frac{1}{2}} \leadsto \color{blue}{\left({x}^2 \cdot \frac{9}{40} - \frac{27}{2800} \cdot {x}^{4}\right) - \frac{1}{2}}\]
      0.0
  1. Removed slow pow expressions

Original test:


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