Derivation

Split input into 2 regimes
if x < -0.028805839143768792 or 0.02136162486376371 < 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. Taylor expanded around inf 0.0
  \[\leadsto \frac{\color{blue}{x - \sin x}}{x - \tan x}\]
4. Using strategy rm
5. Applied log1p-expm1-u0.0
  \[\leadsto \color{blue}{\log_* (1 + (e^{\frac{x - \sin x}{x - \tan x}} - 1)^*)}\]
if -0.028805839143768792 < x < 0.02136162486376371
1. Initial program 62.8
  \[\frac{x - \sin x}{x - \tan x}\]
2. Initial simplification62.8
  \[\leadsto \frac{x - \sin x}{x - \tan x}\]
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
  \[\leadsto \color{blue}{\left(\frac{9}{40} \cdot x\right) \cdot x - (\left({x}^{4}\right) \cdot \frac{27}{2800} + \frac{1}{2})_*}\]
Recombined 2 regimes into one program.
Final simplification0.0
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.028805839143768792 \lor \neg \left(x \le 0.02136162486376371\right):\\ \;\;\;\;\log_* (1 + (e^{\frac{x - \sin x}{x - \tan x}} - 1)^*)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{9}{40} \cdot x\right) \cdot x - (\left({x}^{4}\right) \cdot \frac{27}{2800} + \frac{1}{2})_*\\ \end{array}\]

Runtime

herbie shell --seed 2018234 +o rules:numerics
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))