Derivation

Split input into 2 regimes
if x < -0.024607759771543385 or 0.03109396399534124 < x
1. Initial program 0.0
  \[\frac{x - \sin x}{x - \tan x}\]
2. Initial simplification0.0
  \[\leadsto \frac{x - \sin x}{x - \tan x}\]
3. Using strategy rm
4. Applied div-sub0.0
  \[\leadsto \color{blue}{\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}}\]
if -0.024607759771543385 < x < 0.03109396399534124
1. Initial program 62.7
  \[\frac{x - \sin x}{x - \tan x}\]
2. Initial simplification62.7
  \[\leadsto \frac{x - \sin x}{x - \tan x}\]
3. Taylor expanded around 0 0.0
  \[\leadsto \color{blue}{\frac{9}{40} \cdot {x}^{2} - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)}\]
4. Simplified0.0
  \[\leadsto \color{blue}{(\left(\frac{9}{40} \cdot x\right) \cdot x + \left((\frac{-27}{2800} \cdot \left({x}^{4}\right) + \frac{-1}{2})_*\right))_*}\]
5. Taylor expanded around -inf 0.0
  \[\leadsto (\color{blue}{\left(\frac{9}{40} \cdot x\right)} \cdot x + \left((\frac{-27}{2800} \cdot \left({x}^{4}\right) + \frac{-1}{2})_*\right))_*\]
Recombined 2 regimes into one program.
Final simplification0.0
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.024607759771543385 \lor \neg \left(x \le 0.03109396399534124\right):\\ \;\;\;\;\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{9}{40} \cdot x\right) \cdot x + \left((\frac{-27}{2800} \cdot \left({x}^{4}\right) + \frac{-1}{2})_*\right))_*\\ \end{array}\]

Runtime

herbie shell --seed 2018348 +o rules:numerics
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))