Error

Derivation

1. Split input into 2 regimes

2. if x < -2.503034322186462 or 2.4435472739275776 < x

   1. Initial program 0.0

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

   2. Taylor expanded around -inf 0.5

      \[\leadsto \color{blue}{\left(1 + \left(\frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2} \cdot {x}^{2}} + \frac{\sin x}{\cos x \cdot x}\right)\right) - \left(\frac{\sin x}{x} + \frac{{\left(\sin x\right)}^{2}}{\cos x \cdot {x}^{2}}\right)}\]

   if -2.503034322186462 < x < 2.4435472739275776

   1. Initial program 62.4

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

   2. Taylor expanded around 0 0.3

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

   3. Simplified0.3

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

4. Final simplification0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.503034322186462 \lor \neg \left(x \le 2.4435472739275776\right):\\ \;\;\;\;\left(\left(\frac{\sin x}{\cos x \cdot x} + \frac{{\left(\sin x\right)}^{2}}{{x}^{2} \cdot {\left(\cos x\right)}^{2}}\right) + 1\right) - \left(\frac{{\left(\sin x\right)}^{2}}{{x}^{2} \cdot \cos x} + \frac{\sin x}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{9}{40} \cdot x\right) \cdot x + \left((\frac{-27}{2800} \cdot \left({x}^{4}\right) + \frac{-1}{2})_*\right))_*\\ \end{array}\]

Runtime

Time bar (total: 48.0s)Debug logProfile

herbie shell --seed 2018254 +o rules:numerics
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))