\[\frac{x - \sin x}{x - \tan x}\]
Test:
NMSE problem 3.4.5
Bits:
128 bits
Bits error versus x
Time: 22.9 s
Input Error: 31.7
Output Error: 0.2
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x} & \text{when } x \le -1.7643470284436324 \cdot 10^{-06} \\ \log \left(\frac{e^{\frac{9}{40} \cdot {x}^2}}{e^{\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}}}\right) & \text{when } x \le 5.107891217442523 \cdot 10^{+18} \\ \frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x} & \text{otherwise} \end{cases}\)

    if x < -1.7643470284436324e-06 or 5.107891217442523e+18 < x

    1. Started with
      \[\frac{x - \sin x}{x - \tan x}\]
      0.2
    2. Using strategy rm
      0.2
    3. Applied div-sub to get
      \[\color{red}{\frac{x - \sin x}{x - \tan x}} \leadsto \color{blue}{\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}}\]
      0.2

    if -1.7643470284436324e-06 < x < 5.107891217442523e+18

    1. Started with
      \[\frac{x - \sin x}{x - \tan x}\]
      62.9
    2. Applied taylor to get
      \[\frac{x - \sin x}{x - \tan x} \leadsto \frac{9}{40} \cdot {x}^2 - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)\]
      0.0
    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. Using strategy rm
      0.0
    5. Applied add-log-exp to get
      \[\frac{9}{40} \cdot {x}^2 - \color{red}{\left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)} \leadsto \frac{9}{40} \cdot {x}^2 - \color{blue}{\log \left(e^{\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}}\right)}\]
      0.2
    6. Applied add-log-exp to get
      \[\color{red}{\frac{9}{40} \cdot {x}^2} - \log \left(e^{\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}}\right) \leadsto \color{blue}{\log \left(e^{\frac{9}{40} \cdot {x}^2}\right)} - \log \left(e^{\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}}\right)\]
      0.2
    7. Applied diff-log to get
      \[\color{red}{\log \left(e^{\frac{9}{40} \cdot {x}^2}\right) - \log \left(e^{\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}}\right)} \leadsto \color{blue}{\log \left(\frac{e^{\frac{9}{40} \cdot {x}^2}}{e^{\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}}}\right)}\]
      0.2
  1. Removed slow pow expressions

Original test:


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