Derivation

Split input into 2 regimes
if x < -0.027452856968842716 or 0.027301495071021288 < x
1. Initial program 0.0
  \[\frac{x - \sin x}{x - \tan x}\]
2. Using strategy rm
3. Applied div-sub0.0
  \[\leadsto \color{blue}{\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}}\]
if -0.027452856968842716 < x < 0.027301495071021288
1. Initial program 62.8
  \[\frac{x - \sin x}{x - \tan x}\]
2. Taylor expanded around 0 0.0
  \[\leadsto \color{blue}{\frac{9}{40} \cdot {x}^{2} - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)}\]
3. Simplified0.0
  \[\leadsto \color{blue}{\left(\frac{9}{40} \cdot x\right) \cdot x - (\left({x}^{4}\right) \cdot \frac{27}{2800} + \frac{1}{2})_*}\]
4. Using strategy rm
5. Applied fma-udef0.0
  \[\leadsto \left(\frac{9}{40} \cdot x\right) \cdot x - \color{blue}{\left({x}^{4} \cdot \frac{27}{2800} + \frac{1}{2}\right)}\]
6. Applied associate--r+0.0
  \[\leadsto \color{blue}{\left(\left(\frac{9}{40} \cdot x\right) \cdot x - {x}^{4} \cdot \frac{27}{2800}\right) - \frac{1}{2}}\]
Recombined 2 regimes into one program.
Final simplification0.0
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.027452856968842716 \lor \neg \left(x \le 0.027301495071021288\right):\\ \;\;\;\;\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{9}{40} \cdot x\right) \cdot x - {x}^{4} \cdot \frac{27}{2800}\right) - \frac{1}{2}\\ \end{array}\]

Runtime

herbie shell --seed 2018217 +o rules:numerics
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))

In
Out