Derivation

Split input into 2 regimes
if x < -0.02532179266822333 or 0.027814961976684326 < x
1. Initial program 0.1
  \[\frac{x - \sin x}{x - \tan x}\]
2. Using strategy rm
3. Applied div-sub0.1
  \[\leadsto \color{blue}{\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}}\]
if -0.02532179266822333 < x < 0.027814961976684326
1. Initial program 62.9
  \[\frac{x - \sin x}{x - \tan x}\]
2. Taylor expanded around 0 0.0
  \[\leadsto \color{blue}{\frac{9}{40} \cdot {x}^{2} - \left(\frac{1}{2} + \frac{27}{2800} \cdot {x}^{4}\right)}\]
3. Applied simplify0.0
  \[\leadsto \color{blue}{x \cdot \left(x \cdot \frac{9}{40}\right) - (\frac{27}{2800} \cdot \left({x}^{4}\right) + \frac{1}{2})_*}\]
4. Using strategy rm
5. Applied log1p-expm1-u0.0
  \[\leadsto \color{blue}{\log_* (1 + (e^{x \cdot \left(x \cdot \frac{9}{40}\right) - (\frac{27}{2800} \cdot \left({x}^{4}\right) + \frac{1}{2})_*} - 1)^*)}\]
Recombined 2 regimes into one program.
Applied simplify0.0
\[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;x \le -0.02532179266822333 \lor \neg \left(x \le 0.027814961976684326\right):\\ \;\;\;\;\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\\ \mathbf{else}:\\ \;\;\;\;\log_* (1 + (e^{x \cdot \left(x \cdot \frac{9}{40}\right) - (\frac{27}{2800} \cdot \left({x}^{4}\right) + \frac{1}{2})_*} - 1)^*)\\ \end{array}}\]

Runtime

herbie shell --seed 2018198 +o rules:numerics
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))