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Derivation

1. Split input into 2 regimes

2. if x < -0.028126907832324227 or 0.029951952789081897 < x

   1. Initial program 0.0

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

   2. Initial simplification0.0

      \[\leadsto \frac{x - \sin x}{x - \tan x}\]
   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-neg0.1

      \[\leadsto \color{blue}{\frac{x}{x - \tan x} + \left(-\frac{\sin x}{x - \tan x}\right)}\]

   if -0.028126907832324227 < x < 0.029951952789081897

   1. Initial program 62.7

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

   2. Initial simplification62.7

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

   4. Applied div-sub62.6

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

   6. Applied sub-neg62.6

      \[\leadsto \color{blue}{\frac{x}{x - \tan x} + \left(-\frac{\sin x}{x - \tan x}\right)}\]

   7. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{\frac{9}{40} \cdot {x}^{2} - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.028126907832324227 \lor \neg \left(x \le 0.029951952789081897\right):\\ \;\;\;\;\frac{x}{x - \tan x} + \frac{-\sin x}{x - \tan x}\\ \mathbf{else}:\\ \;\;\;\;\frac{9}{40} \cdot {x}^{2} - \left(\frac{1}{2} + \frac{27}{2800} \cdot {x}^{4}\right)\\ \end{array}\]

Runtime

Time bar (total: 37.8s)Debug logProfile

herbie shell --seed 2018256 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))