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Derivation

1. Split input into 3 regimes

2. if x < -1.5981192504292256

   1. Initial program 0.0

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

   3. Applied add-sqr-sqrt0.1

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

   if -1.5981192504292256 < x < 0.03156574465888638

   1. Initial program 62.7

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

   2. Taylor expanded around 0 0.1

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

   4. Applied associate--r+0.1

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

   if 0.03156574465888638 < x

   1. Initial program 0.1

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

   3. Applied div-sub0.1

      \[\leadsto \color{blue}{\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}}\]
3. Recombined 3 regimes into one program.

4. Final simplification0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.5981192504292256:\\ \;\;\;\;\sqrt{\frac{x - \sin x}{x - \tan x}} \cdot \sqrt{\frac{x - \sin x}{x - \tan x}}\\ \mathbf{elif}\;x \le 0.03156574465888638:\\ \;\;\;\;\left(\frac{9}{40} \cdot {x}^{2} - \frac{27}{2800} \cdot {x}^{4}\right) - \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019005 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))

Details

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