Derivation

Split input into 2 regimes
if x < -0.017033296873819756 or 0.016375988937567955 < x
1. Initial program 0.0
  \[\frac{x - \sin x}{x - \tan x}\]
2. Using strategy rm
3. Applied expm1-log1p-u0.1
  \[\leadsto \color{blue}{(e^{\log_* (1 + \frac{x - \sin x}{x - \tan x})} - 1)^*}\]
if -0.017033296873819756 < x < 0.016375988937567955
1. Initial program 62.9
  \[\frac{x - \sin x}{x - \tan x}\]
2. Using strategy rm
3. Applied expm1-log1p-u62.9
  \[\leadsto \color{blue}{(e^{\log_* (1 + \frac{x - \sin x}{x - \tan x})} - 1)^*}\]
4. Taylor expanded around 0 0.0
  \[\leadsto (e^{\color{blue}{\left(\log \frac{1}{2} + \frac{9}{20} \cdot {x}^{2}\right) - \frac{27}{224} \cdot {x}^{4}}} - 1)^*\]
5. Simplified0.0
  \[\leadsto (e^{\color{blue}{(\left(x \cdot x\right) \cdot \frac{9}{20} + \left(\log \frac{1}{2}\right))_* - \frac{27}{224} \cdot {x}^{4}}} - 1)^*\]
Recombined 2 regimes into one program.
Final simplification0.0
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.017033296873819756 \lor \neg \left(x \le 0.016375988937567955\right):\\ \;\;\;\;(e^{\log_* (1 + \frac{x - \sin x}{x - \tan x})} - 1)^*\\ \mathbf{else}:\\ \;\;\;\;(e^{(\left(x \cdot x\right) \cdot \frac{9}{20} + \left(\log \frac{1}{2}\right))_* - \frac{27}{224} \cdot {x}^{4}} - 1)^*\\ \end{array}\]

Runtime

herbie shell --seed 2018219 +o rules:numerics
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))