\[\frac{x - \sin x}{x - \tan x}\]
Test:
NMSE problem 3.4.5
Bits:
128 bits
Bits error versus x
Time: 20.3 s
Input Error: 31.7
Output Error: 0.0
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{x - \sin x}{x - \tan x} & \text{when } x \le -0.0003019673949516488 \\ \frac{9}{40} \cdot {x}^2 - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right) & \text{when } x \le 36050.53017192879 \\ \frac{x - \sin x}{x - \tan x} & \text{otherwise} \end{cases}\)

    if x < -0.0003019673949516488 or 36050.53017192879 < x

    1. Started with
      \[\frac{x - \sin x}{x - \tan x}\]
      0.1

    if -0.0003019673949516488 < x < 36050.53017192879

    1. Started with
      \[\frac{x - \sin x}{x - \tan x}\]
      62.8
    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
  1. Removed slow pow expressions

Original test:


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