Error

Derivation

1. Split input into 2 regimes

2. if x < -0.026359463711262344 or 0.03208515368603683 < x

   1. Initial program 0.0

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

   2. Taylor expanded around -inf 0.0

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

   if -0.026359463711262344 < x < 0.03208515368603683

   1. Initial program 62.9

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

   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

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

   4. Taylor expanded around 0 0.0

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

   5. Simplified0.0

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

4. Final simplification0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.026359463711262344 \lor \neg \left(x \le 0.03208515368603683\right):\\ \;\;\;\;\frac{x - \sin x}{x - \frac{\sin x}{\cos x}}\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{9}{40} \cdot x\right) \cdot x + \left((\frac{-27}{2800} \cdot \left({x}^{4}\right) + \frac{-1}{2})_*\right))_*\\ \end{array}\]

Runtime

Time bar (total: 29.5s)Debug logProfile

herbie shell --seed 2018297 +o rules:numerics
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))