Error

Derivation

1. Split input into 2 regimes

2. if x < -0.02773371216869672 or 0.02992458358046798 < 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 div-sub0.0

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

   if -0.02773371216869672 < x < 0.02992458358046798

   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 inf 62.7

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

   5. Applied div-sub62.7

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

   6. Taylor expanded around 0 0.0

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

   7. Simplified0.0

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

4. Final simplification0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.02773371216869672 \lor \neg \left(x \le 0.02992458358046798\right):\\ \;\;\;\;\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{9}{40} \cdot x\right) \cdot x + \left((\frac{-27}{2800} \cdot \left({x}^{4}\right) + \frac{-1}{2})_*\right))_*\\ \end{array}\]

Runtime

Time bar (total: 1.0m)Debug logProfile

herbie shell --seed 2018286 +o rules:numerics
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))