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Derivation

1. Split input into 2 regimes

2. if x < -0.030793827110981598 or 0.02802111767978534 < 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 add-cbrt-cube0.0

      \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{x - \sin x}{x - \tan x} \cdot \frac{x - \sin x}{x - \tan x}\right) \cdot \frac{x - \sin x}{x - \tan x}}}\]

   if -0.030793827110981598 < x < 0.02802111767978534

   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 0.0

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

   4. Taylor expanded around -inf 63.6

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

   5. Simplified0.0

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

4. Final simplification0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.030793827110981598 \lor \neg \left(x \le 0.02802111767978534\right):\\ \;\;\;\;\sqrt[3]{\frac{x - \sin x}{x - \tan x} \cdot \left(\frac{x - \sin x}{x - \tan x} \cdot \frac{x - \sin x}{x - \tan x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(\frac{9}{40} \cdot x\right) - \frac{27}{2800} \cdot {x}^{4}\right) - \frac{1}{2}\\ \end{array}\]

Runtime

Time bar (total: 40.7s)Debug logProfile

herbie shell --seed 2018225 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))