\[\frac{x - \sin x}{x - \tan x}\]
Test:
NMSE problem 3.4.5
Bits:
128 bits
Bits error versus x
Time: 29.6 s
Input Error: 31.7
Output Error: 0.3
Log: ⚲
Profile: 🕒
\(\begin{cases} \log_* (1 + (e^{\frac{x - \sin x}{x - \tan x}} - 1)^*) & \text{when } x \le -8.61646249982272 \cdot 10^{-12} \\ \frac{9}{40} \cdot {x}^2 - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right) & \text{when } x \le 8.584780819621077 \\ \frac{{\left(\frac{x}{x - \tan x}\right)}^3 - {\left(\frac{\sin x}{x - \tan x}\right)}^3}{(\left(\frac{x}{x - \tan x}\right) * \left(\frac{\sin x}{x - \tan x} + \frac{x}{x - \tan x}\right) + \left({\left(\frac{\sin x}{x - \tan x}\right)}^2\right))_*} & \text{otherwise} \end{cases}\)

    if x < -8.61646249982272e-12

    1. Started with
      \[\frac{x - \sin x}{x - \tan x}\]
      1.0
    2. Using strategy rm
      1.0
    3. Applied log1p-expm1-u to get
      \[\color{red}{\frac{x - \sin x}{x - \tan x}} \leadsto \color{blue}{\log_* (1 + (e^{\frac{x - \sin x}{x - \tan x}} - 1)^*)}\]
      1.0

    if -8.61646249982272e-12 < x < 8.584780819621077

    1. Started with
      \[\frac{x - \sin x}{x - \tan x}\]
      63.0
    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

    if 8.584780819621077 < 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
    4. Using strategy rm
      0.0
    5. Applied flip3-- to get
      \[\color{red}{\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}} \leadsto \color{blue}{\frac{{\left(\frac{x}{x - \tan x}\right)}^{3} - {\left(\frac{\sin x}{x - \tan x}\right)}^{3}}{{\left(\frac{x}{x - \tan x}\right)}^2 + \left({\left(\frac{\sin x}{x - \tan x}\right)}^2 + \frac{x}{x - \tan x} \cdot \frac{\sin x}{x - \tan x}\right)}}\]
      0.0
    6. Applied simplify to get
      \[\frac{\color{red}{{\left(\frac{x}{x - \tan x}\right)}^{3} - {\left(\frac{\sin x}{x - \tan x}\right)}^{3}}}{{\left(\frac{x}{x - \tan x}\right)}^2 + \left({\left(\frac{\sin x}{x - \tan x}\right)}^2 + \frac{x}{x - \tan x} \cdot \frac{\sin x}{x - \tan x}\right)} \leadsto \frac{\color{blue}{{\left(\frac{x}{x - \tan x}\right)}^3 - {\left(\frac{\sin x}{x - \tan x}\right)}^3}}{{\left(\frac{x}{x - \tan x}\right)}^2 + \left({\left(\frac{\sin x}{x - \tan x}\right)}^2 + \frac{x}{x - \tan x} \cdot \frac{\sin x}{x - \tan x}\right)}\]
      0.0
    7. Applied simplify to get
      \[\frac{{\left(\frac{x}{x - \tan x}\right)}^3 - {\left(\frac{\sin x}{x - \tan x}\right)}^3}{\color{red}{{\left(\frac{x}{x - \tan x}\right)}^2 + \left({\left(\frac{\sin x}{x - \tan x}\right)}^2 + \frac{x}{x - \tan x} \cdot \frac{\sin x}{x - \tan x}\right)}} \leadsto \frac{{\left(\frac{x}{x - \tan x}\right)}^3 - {\left(\frac{\sin x}{x - \tan x}\right)}^3}{\color{blue}{(\left(\frac{x}{x - \tan x}\right) * \left(\frac{\sin x}{x - \tan x} + \frac{x}{x - \tan x}\right) + \left({\left(\frac{\sin x}{x - \tan x}\right)}^2\right))_*}}\]
      0.0
  1. Removed slow pow expressions

Original test:


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