Derivation

Split input into 3 regimes
if x < -1.5761806368521758
1. Initial program 0.0
  \[\frac{x - \sin x}{x - \tan x}\]
2. Taylor expanded around -inf 0.0
  \[\leadsto \frac{\color{blue}{x - \sin x}}{x - \tan x}\]
3. Using strategy rm
4. Applied add-sqr-sqrt0.0
  \[\leadsto \color{blue}{\sqrt{\frac{x - \sin x}{x - \tan x}} \cdot \sqrt{\frac{x - \sin x}{x - \tan x}}}\]
if -1.5761806368521758 < x < 0.028892187182056106
1. Initial program 62.4
  \[\frac{x - \sin x}{x - \tan x}\]
2. Taylor expanded around 0 0.1
  \[\leadsto \color{blue}{\frac{9}{40} \cdot {x}^{2} - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)}\]
3. Simplified0.1
  \[\leadsto \color{blue}{(\left(\frac{9}{40} \cdot x\right) \cdot x + \left((\frac{-27}{2800} \cdot \left({x}^{4}\right) + \frac{-1}{2})_*\right))_*}\]
if 0.028892187182056106 < x
1. Initial program 0.0
  \[\frac{x - \sin x}{x - \tan x}\]
2. Taylor expanded around -inf 0.0
  \[\leadsto \frac{\color{blue}{x - \sin x}}{x - \tan x}\]
Recombined 3 regimes into one program.
Final simplification0.1
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.5761806368521758:\\ \;\;\;\;\sqrt{\frac{x - \sin x}{x - \tan x}} \cdot \sqrt{\frac{x - \sin x}{x - \tan x}}\\ \mathbf{elif}\;x \le 0.028892187182056106:\\ \;\;\;\;(\left(\frac{9}{40} \cdot x\right) \cdot x + \left((\frac{-27}{2800} \cdot \left({x}^{4}\right) + \frac{-1}{2})_*\right))_*\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \end{array}\]

Runtime

herbie shell --seed 2018252 +o rules:numerics
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))