Derivation

Split input into 2 regimes
if x < -0.029538973901007348 or 0.03088865576084561 < x
1. Initial program 0.1
  \[\frac{x - \sin x}{x - \tan x}\]
2. Using strategy rm
3. Applied div-sub0.1
  \[\leadsto \color{blue}{\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}}\]
if -0.029538973901007348 < x < 0.03088865576084561
1. Initial program 62.6
  \[\frac{x - \sin x}{x - \tan x}\]
2. Using strategy rm
3. Applied div-sub62.6
  \[\leadsto \color{blue}{\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}}\]
4. Taylor expanded around 0 0.0
  \[\leadsto \color{blue}{\frac{9}{40} \cdot {x}^{2} - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)}\]
5. Simplified0.0
  \[\leadsto \color{blue}{\left(\frac{9}{40} \cdot x\right) \cdot x - (\left({x}^{4}\right) \cdot \frac{27}{2800} + \frac{1}{2})_*}\]
Recombined 2 regimes into one program.
Final simplification0.0
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.029538973901007348 \lor \neg \left(x \le 0.03088865576084561\right):\\ \;\;\;\;\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{9}{40} \cdot x\right) \cdot x - (\left({x}^{4}\right) \cdot \frac{27}{2800} + \frac{1}{2})_*\\ \end{array}\]

Runtime

herbie shell --seed 2018230 +o rules:numerics
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))