Derivation

Split input into 2 regimes
if x < -0.026359463711262344 or 0.03208515368603683 < 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
  \[\leadsto \frac{x - \sin x}{\color{blue}{x - \frac{\sin x}{\cos x}}}\]
if -0.026359463711262344 < x < 0.03208515368603683
1. Initial program 62.9
  \[\frac{x - \sin x}{x - \tan x}\]
2. Initial simplification62.9
  \[\leadsto \frac{x - \sin x}{x - \tan x}\]
3. Taylor expanded around -inf 62.9
  \[\leadsto \frac{x - \sin x}{\color{blue}{x - \frac{\sin x}{\cos x}}}\]
4. Taylor expanded around 0 0.0
  \[\leadsto \color{blue}{\frac{9}{40} \cdot {x}^{2} - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)}\]
5. Using strategy rm
6. Applied add-log-exp0.0
  \[\leadsto \color{blue}{\log \left(e^{\frac{9}{40} \cdot {x}^{2} - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)}\right)}\]
Recombined 2 regimes into one program.
Final simplification0.0
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.026359463711262344 \lor \neg \left(x \le 0.03208515368603683\right):\\ \;\;\;\;\frac{x - \sin x}{x - \frac{\sin x}{\cos x}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{9}{40} \cdot {x}^{2} - \left(\frac{1}{2} + \frac{27}{2800} \cdot {x}^{4}\right)}\right)\\ \end{array}\]

Runtime

	Baseline	Herbie	Oracle	Span	%
Regimes	31.1	0.0	0.0	31.1	100%

herbie shell --seed 2018297 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))

In
Out