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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -0.03039944257282741:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \mathbf{if}\;x \le 0.024772907566964195:\\ \;\;\;\;\left(x \cdot x\right) \cdot \frac{9}{40} - (\frac{27}{2800} \cdot \left({x}^{4}\right) + \frac{1}{2})_*\\ \mathbf{else}:\\ \;\;\;\;(e^{\log_* (1 + \frac{x - \sin x}{x - \tan x})} - 1)^*\\ \end{array}\]

Derivation

Split input into 3 regimes
if x < -0.03039944257282741
1. Initial program 0.0
  \[\frac{x - \sin x}{x - \tan x}\]
if -0.03039944257282741 < x < 0.024772907566964195
1. Initial program 62.8
  \[\frac{x - \sin x}{x - \tan x}\]
2. Taylor expanded around 0 0.0
  \[\leadsto \color{blue}{\frac{9}{40} \cdot {x}^{2} - \left(\frac{1}{2} + \frac{27}{2800} \cdot {x}^{4}\right)}\]
3. Applied simplify0.0
  \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \frac{9}{40} - (\frac{27}{2800} \cdot \left({x}^{4}\right) + \frac{1}{2})_*}\]
if 0.024772907566964195 < x
1. Initial program 0.1
  \[\frac{x - \sin x}{x - \tan x}\]
2. Using strategy rm
3. Applied expm1-log1p-u0.1
  \[\leadsto \color{blue}{(e^{\log_* (1 + \frac{x - \sin x}{x - \tan x})} - 1)^*}\]
Recombined 3 regimes into one program.

Runtime

herbie shell --seed '#(1064397287 3527694221 3797617954 1138343853 2854031332 1153838279)' +o rules:numerics
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))

Error

Derivation

`if x < -0.03039944257282741`

`if -0.03039944257282741 < x < 0.024772907566964195`

`if 0.024772907566964195 < x`

Runtime