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Derivation

1. Split input into 3 regimes

2. if x < -0.025359436115173863

   1. Initial program 0.1

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

   2. Initial simplification0.1

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

   4. Applied div-sub0.1

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

   if -0.025359436115173863 < x < 0.02997657493859274

   1. Initial program 62.9

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

   2. Initial simplification62.9

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

   3. Taylor expanded around 0 0.0

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

   if 0.02997657493859274 < x

   1. Initial program 0.1

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

   2. Initial simplification0.1

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

   4. Applied div-sub0.1

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

   6. Applied sub-div0.1

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

4. Final simplification0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.025359436115173863:\\ \;\;\;\;\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\\ \mathbf{elif}\;x \le 0.02997657493859274:\\ \;\;\;\;\frac{9}{40} \cdot {x}^{2} - \left(\frac{1}{2} + \frac{27}{2800} \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \end{array}\]

Runtime

Time bar (total: 29.6s)Debug logProfile

herbie shell --seed 2018355 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))