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

⬇

\[\begin{array}{l} \mathbf{if}\;x \le -5.5164922447739564 \cdot 10^{-14}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{x - \sin x}}{\sqrt[3]{x - \tan x}}\right)}^3\\ \mathbf{if}\;x \le 0.22138222595296528:\\ \;\;\;\;\frac{9}{40} \cdot {x}^2 - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{x - \sin x}}{\sqrt[3]{x - \tan x}}\right)}^3\\ \end{array}\]

Derivation

Split input into 2 regimes.
if x < -5.5164922447739564e-14 or 0.22138222595296528 < x
1. Initial program 0.9
  \[\frac{x - \sin x}{x - \tan x}\]
2. Using strategy rm
3. Applied add-cube-cbrt 2.2
  \[\leadsto \frac{x - \sin x}{\color{blue}{{\left(\sqrt[3]{x - \tan x}\right)}^3}}\]
4. Applied add-cube-cbrt 0.9
  \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{x - \sin x}\right)}^3}}{{\left(\sqrt[3]{x - \tan x}\right)}^3}\]
5. Applied cube-undiv 0.9
  \[\leadsto \color{blue}{{\left(\frac{\sqrt[3]{x - \sin x}}{\sqrt[3]{x - \tan x}}\right)}^3}\]
if -5.5164922447739564e-14 < x < 0.22138222595296528
1. Initial program 63.0
  \[\frac{x - \sin x}{x - \tan x}\]
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 '#(1064651971 495577305 2200811460 13024471 864198081 231948279)'
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))

Error

Derivation

`if x < -5.5164922447739564e-14 or 0.22138222595296528 < x`

`if -5.5164922447739564e-14 < x < 0.22138222595296528`

Runtime