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Derivation

1. Split input into 2 regimes

2. if x < -0.028791956980749166 or 0.02637643178080528 < 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-sub0.0

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

   if -0.028791956980749166 < x < 0.02637643178080528

   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. Using strategy rm

   4. Applied div-sub62.8

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

   5. Taylor expanded around 0 0.0

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

   7. Applied associate--r+0.0

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

4. Final simplification0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.028791956980749166 \lor \neg \left(x \le 0.02637643178080528\right):\\ \;\;\;\;\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{9}{40} \cdot {x}^{2} - \frac{27}{2800} \cdot {x}^{4}\right) - \frac{1}{2}\\ \end{array}\]

Runtime

Time bar (total: 27.1s)Debug logProfile

herbie shell --seed 2018353 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))