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

⬇

\[\begin{array}{l} \mathbf{if}\;x \le -5.458497096453193 \cdot 10^{-06}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{x - \sin x}{x - \tan x}\right)}^3}\\ \mathbf{if}\;x \le 85.36504497937187:\\ \;\;\;\;\frac{9}{40} \cdot {x}^2 - \left(\frac{1}{2} + \frac{27}{2800} \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{x - \sin x}{x - \tan x}\right)}^3}\\ \end{array}\]

Derivation

Split input into 2 regimes.
if x < -5.458497096453193e-06 or 85.36504497937187 < x
1. Initial program 0.2
  \[\frac{x - \sin x}{x - \tan x}\]
2. Using strategy rm
3. Applied add-cbrt-cube 41.6
  \[\leadsto \frac{x - \sin x}{\color{blue}{\sqrt[3]{{\left(x - \tan x\right)}^3}}}\]
4. Applied add-cbrt-cube 41.5
  \[\leadsto \frac{\color{blue}{\sqrt[3]{{\left(x - \sin x\right)}^3}}}{\sqrt[3]{{\left(x - \tan x\right)}^3}}\]
5. Applied cbrt-undiv 41.4
  \[\leadsto \color{blue}{\sqrt[3]{\frac{{\left(x - \sin x\right)}^3}{{\left(x - \tan x\right)}^3}}}\]
6. Applied simplify 0.2
  \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{x - \sin x}{x - \tan x}\right)}^3}}\]
if -5.458497096453193e-06 < x < 85.36504497937187
1. Initial program 62.9
  \[\frac{x - \sin x}{x - \tan x}\]
2. Applied taylor 0.0
  \[\leadsto \frac{9}{40} \cdot {x}^2 - \left(\frac{1}{2} + \frac{27}{2800} \cdot {x}^{4}\right)\]
3. Taylor expanded around 0 0.0
  
  \[\leadsto \color{blue}{\frac{9}{40} \cdot {x}^2 - \left(\frac{1}{2} + \frac{27}{2800} \cdot {x}^{4}\right)}\]
Recombined 2 regimes into one program.
Removed slow pow expressions

Runtime

Please include this information when filing a bug report:

herbie --seed '#(1221977421 1476470008 2397091756 2667547678 102625347 3788899007)'
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))

Error

Derivation

`if x < -5.458497096453193e-06 or 85.36504497937187 < x`

`if -5.458497096453193e-06 < x < 85.36504497937187`

Runtime