Error

Derivation

1. Split input into 3 regimes

2. if x < -0.02864047107391261

   1. Initial program 0.0

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

   2. Taylor expanded around -inf 0.0

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

   4. Applied clear-num0.0

      \[\leadsto \color{blue}{\frac{1}{\frac{x - \tan x}{x - \sin x}}}\]

   if -0.02864047107391261 < x < 0.028319229094880563

   1. Initial program 62.7

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

   2. Taylor expanded around -inf 62.7

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

   4. Applied clear-num62.7

      \[\leadsto \color{blue}{\frac{1}{\frac{x - \tan x}{x - \sin x}}}\]

   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. Simplified0.0

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

   if 0.028319229094880563 < x

   1. Initial program 0.1

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

   2. Taylor expanded around -inf 0.1

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

4. Final simplification0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.02864047107391261:\\ \;\;\;\;\frac{1}{\frac{x - \tan x}{x - \sin x}}\\ \mathbf{elif}\;x \le 0.028319229094880563:\\ \;\;\;\;\frac{9}{40} \cdot \left(x \cdot x\right) - \left({x}^{4} \cdot \frac{27}{2800} - \frac{-1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019093 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))