\[\frac{x - \sin x}{x - \tan x}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le -0.03524652167209293:\\ \;\;\;\;\log \left(e^{\frac{x - \sin x}{x - \tan x}}\right)\\ \mathbf{if}\;x \le 0.03196978990602315:\\ \;\;\;\;x \cdot \left(x \cdot \frac{9}{40}\right) - (\frac{27}{2800} \cdot \left({x}^{4}\right) + \frac{1}{2})_*\\ \mathbf{else}:\\ \;\;\;\;\log_* (1 + (e^{\frac{x - \sin x}{x - \tan x}} - 1)^*)\\ \end{array}\]

Derivation

Split input into 3 regimes
if x < -0.03524652167209293
1. Initial program 0.0
  \[\frac{x - \sin x}{x - \tan x}\]
2. Using strategy rm
3. Applied add-log-exp0.0
  \[\leadsto \color{blue}{\log \left(e^{\frac{x - \sin x}{x - \tan x}}\right)}\]
if -0.03524652167209293 < x < 0.03196978990602315
1. Initial program 62.7
  \[\frac{x - \sin x}{x - \tan x}\]
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
  \[\leadsto \color{blue}{x \cdot \left(x \cdot \frac{9}{40}\right) - (\frac{27}{2800} \cdot \left({x}^{4}\right) + \frac{1}{2})_*}\]
if 0.03196978990602315 < x
1. Initial program 0.0
  \[\frac{x - \sin x}{x - \tan x}\]
2. Using strategy rm
3. Applied log1p-expm1-u0.0
  \[\leadsto \color{blue}{\log_* (1 + (e^{\frac{x - \sin x}{x - \tan x}} - 1)^*)}\]
Recombined 3 regimes into one program.

Runtime

herbie shell --seed '#(1070609872 3456127585 2380521889 2328837196 1765472538 734540918)' +o rules:numerics
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))

Error

Derivation

`if x < -0.03524652167209293`

`if -0.03524652167209293 < x < 0.03196978990602315`

`if 0.03196978990602315 < x`

Runtime