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Derivation

1. Split input into 3 regimes

2. if x < -0.02783012588450388

   1. Initial program 0.0

      \[\frac{x - \sin x}{x - \tan x}\]
   2. Using strategy rm

   3. Applied div-sub0.0

      \[\leadsto \color{blue}{\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}}\]

   if -0.02783012588450388 < x < 0.028560118651481507

   1. Initial program 62.8

      \[\frac{x - \sin x}{x - \tan x}\]
   2. Using strategy rm

   3. Applied div-sub62.7

      \[\leadsto \color{blue}{\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}}\]

   4. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{\frac{9}{40} \cdot {x}^{2} - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)}\]
   5. Using strategy rm

   6. Applied associate--r+0.0

      \[\leadsto \color{blue}{\left(\frac{9}{40} \cdot {x}^{2} - \frac{27}{2800} \cdot {x}^{4}\right) - \frac{1}{2}}\]

   if 0.028560118651481507 < x

   1. Initial program 0.1

      \[\frac{x - \sin x}{x - \tan x}\]
   2. Using strategy rm

   3. Applied div-sub0.0

      \[\leadsto \color{blue}{\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}}\]
   4. Using strategy rm

   5. Applied add-cbrt-cube0.1

      \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\right) \cdot \left(\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\right)\right) \cdot \left(\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\right)}}\]
3. Recombined 3 regimes into one program.

4. Final simplification0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.02783012588450388:\\ \;\;\;\;\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\\ \mathbf{elif}\;x \le 0.028560118651481507:\\ \;\;\;\;\left(\frac{9}{40} \cdot {x}^{2} - \frac{27}{2800} \cdot {x}^{4}\right) - \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\right) \cdot \left(\left(\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\right) \cdot \left(\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019004 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))

Details

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