Error

Derivation

1. Split input into 2 regimes

2. if x < -0.030979862748587897 or 0.028462903907912153 < x

   1. Initial program 0.0

      \[\frac{x - \sin x}{x - \tan x}\]
   2. Using strategy rm

   3. Applied div-sub0.0

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

   if -0.030979862748587897 < x < 0.028462903907912153

   1. Initial program 62.7

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

   2. Taylor expanded around 0 0.0

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

   3. Simplified0.0

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

   5. Applied sub-neg0.0

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

   6. Applied associate--r+0.0

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

   7. Simplified0.0

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

4. Final simplification0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.030979862748587897:\\ \;\;\;\;\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\\ \mathbf{elif}\;x \le 0.028462903907912153:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot \frac{9}{40} - {x}^{4} \cdot \frac{27}{2800}\right) - \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019050 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))