Derivation

Split input into 2 regimes
if x < -0.02789661879569553 or 0.027376994180579275 < 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 clear-num0.0
  \[\leadsto \color{blue}{\frac{1}{\frac{x - \tan x}{x - \sin x}}}\]
5. Using strategy rm
6. Applied log1p-expm1-u0.1
  \[\leadsto \frac{1}{\color{blue}{\log_* (1 + (e^{\frac{x - \tan x}{x - \sin x}} - 1)^*)}}\]
if -0.02789661879569553 < x < 0.027376994180579275
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
  \[\leadsto \color{blue}{(\left(\frac{9}{40} \cdot x\right) \cdot x + \left((\frac{-27}{2800} \cdot \left({x}^{4}\right) + \frac{-1}{2})_*\right))_*}\]
Recombined 2 regimes into one program.
Final simplification0.0
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.02789661879569553 \lor \neg \left(x \le 0.027376994180579275\right):\\ \;\;\;\;\frac{1}{\log_* (1 + (e^{\frac{x - \tan x}{x - \sin x}} - 1)^*)}\\ \mathbf{else}:\\ \;\;\;\;(\left(x \cdot \frac{9}{40}\right) \cdot x + \left((\frac{-27}{2800} \cdot \left({x}^{4}\right) + \frac{-1}{2})_*\right))_*\\ \end{array}\]

Runtime

herbie shell --seed 2018255 +o rules:numerics
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))