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Derivation

1. Split input into 2 regimes

2. if (tan x) < -0.011256180942606888 or 2.647695068660539e-11 < (tan x)

   1. Initial program 0.5

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

   2. Initial simplification0.5

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

   3. Taylor expanded around inf 0.5

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

   if -0.011256180942606888 < (tan x) < 2.647695068660539e-11

   1. Initial program 63.0

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

   2. Initial simplification63.0

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

   3. Taylor expanded around inf 63.0

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

   4. Taylor expanded around 0 0.2

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

4. Final simplification0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan x \le -0.011256180942606888 \lor \neg \left(\tan x \le 2.647695068660539 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \mathbf{else}:\\ \;\;\;\;{x}^{2} \cdot \frac{9}{40} - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)\\ \end{array}\]

Runtime

Time bar (total: 1.3m)Debug logProfile

herbie shell --seed 2018277 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))