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Derivation

1. Split input into 3 regimes

2. if x < -0.029602678921785613

   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. Taylor expanded around -inf 0.0

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

   if -0.029602678921785613 < x < 1.5604503921256918

   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. Taylor expanded around 0 0.1

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

   5. 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 1.5604503921256918 < 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. Taylor expanded around -inf 0.0

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

   5. Applied add-sqr-sqrt0.0

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

4. Final simplification0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.029602678921785613:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \mathbf{elif}\;x \le 1.5604503921256918:\\ \;\;\;\;\left({x}^{2} \cdot \frac{9}{40} - \frac{27}{2800} \cdot {x}^{4}\right) - \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x - \sin x}{x - \tan x}} \cdot \sqrt{\frac{x - \sin x}{x - \tan x}}\\ \end{array}\]

Runtime

Time bar (total: 27.2s)Debug logProfile

herbie shell --seed 2018290 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))