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

↓

\[\begin{array}{l} \mathbf{if}\;\tan x \le -0.011256180942606888:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \mathbf{elif}\;\tan x \le 2.647695068660539 \cdot 10^{-11}:\\ \;\;\;\;(\left(x \cdot \frac{9}{40}\right) \cdot x + \left((\frac{-27}{2800} \cdot \left({x}^{4}\right) + \frac{-1}{2})_*\right))_*\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \end{array}\]

Derivation

Split input into 2 regimes
if (tan x) < -0.011256180942606888 or 2.647695068660539e-11 < (tan x)
1. Initial program 0.5
  \[\frac{x - \sin x}{x - \tan x}\]
2. Initial simplification0.5
  \[\leadsto \frac{x - \sin x}{x - \tan x}\]
3. Taylor expanded around inf 0.5
  \[\leadsto \frac{\color{blue}{x - \sin x}}{x - \tan x}\]
if -0.011256180942606888 < (tan x) < 2.647695068660539e-11
1. Initial program 63.0
  \[\frac{x - \sin x}{x - \tan x}\]
2. Initial simplification63.0
  \[\leadsto \frac{x - \sin x}{x - \tan x}\]
3. Taylor expanded around 0 0.2
  \[\leadsto \color{blue}{\frac{9}{40} \cdot {x}^{2} - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)}\]
4. Simplified0.2
  \[\leadsto \color{blue}{(\left(\frac{9}{40} \cdot x\right) \cdot x + \left((\frac{-27}{2800} \cdot \left({x}^{4}\right) + \frac{-1}{2})_*\right))_*}\]
Recombined 2 regimes into one program.
Final simplification0.4
\[\leadsto \begin{array}{l} \mathbf{if}\;\tan x \le -0.011256180942606888:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \mathbf{elif}\;\tan x \le 2.647695068660539 \cdot 10^{-11}:\\ \;\;\;\;(\left(x \cdot \frac{9}{40}\right) \cdot x + \left((\frac{-27}{2800} \cdot \left({x}^{4}\right) + \frac{-1}{2})_*\right))_*\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \end{array}\]

Runtime

herbie shell --seed 2018277 +o rules:numerics
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))

Error

Derivation

`if (tan x) < -0.011256180942606888 or 2.647695068660539e-11 < (tan x)`

`if -0.011256180942606888 < (tan x) < 2.647695068660539e-11`

Runtime