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

⬇

\[\begin{array}{l} \mathbf{if}\;x \le -5.640935934993906 \cdot 10^{-07}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{x - \sin x}{x - \tan x}\right)}^3}\\ \mathbf{if}\;x \le 5.981219390019896:\\ \;\;\;\;\frac{9}{40} \cdot {x}^2 - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{x - \sin x}{x - \tan x}\right)}^3}\\ \end{array}\]

Derivation

Split input into 2 regimes.
if x < -5.640935934993906e-07 or 5.981219390019896 < x
1. Initial program 0.2
  \[\frac{x - \sin x}{x - \tan x}\]
2. Using strategy rm
3. Applied add-cbrt-cube 41.0
  \[\leadsto \frac{x - \sin x}{\color{blue}{\sqrt[3]{{\left(x - \tan x\right)}^3}}}\]
4. Applied add-cbrt-cube 40.8
  \[\leadsto \frac{\color{blue}{\sqrt[3]{{\left(x - \sin x\right)}^3}}}{\sqrt[3]{{\left(x - \tan x\right)}^3}}\]
5. Applied cbrt-undiv 40.8
  \[\leadsto \color{blue}{\sqrt[3]{\frac{{\left(x - \sin x\right)}^3}{{\left(x - \tan x\right)}^3}}}\]
6. Applied simplify 0.2
  \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{x - \sin x}{x - \tan x}\right)}^3}}\]
if -5.640935934993906e-07 < x < 5.981219390019896
1. Initial program 62.9
  \[\frac{x - \sin x}{x - \tan x}\]
2. Applied taylor 0.0
  \[\leadsto \frac{9}{40} \cdot {x}^2 - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)\]
3. Taylor expanded around 0 0.0
  
  \[\leadsto \color{blue}{\frac{9}{40} \cdot {x}^2 - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)}\]
Recombined 2 regimes into one program.
Removed slow pow expressions

Runtime

Please include this information when filing a bug report:

herbie shell --seed '#(1067615470 1817955187 3564058462 2385304812 1026693554 2746013648)'
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))

Error

Derivation

`if x < -5.640935934993906e-07 or 5.981219390019896 < x`

`if -5.640935934993906e-07 < x < 5.981219390019896`

Runtime