Error

Derivation

1. Split input into 2 regimes

2. if x < -0.026821773126514865 or 0.030409209466777495 < x

   1. Initial program 0.1

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

   2. Initial simplification0.1

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

   4. Applied clear-num0.1

      \[\leadsto \color{blue}{\frac{1}{\frac{x - \tan x}{x - \sin x}}}\]
   5. Using strategy rm

   6. Applied add-cube-cbrt0.1

      \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{1}{\frac{x - \tan x}{x - \sin x}}} \cdot \sqrt[3]{\frac{1}{\frac{x - \tan x}{x - \sin x}}}\right) \cdot \sqrt[3]{\frac{1}{\frac{x - \tan x}{x - \sin x}}}}\]

   if -0.026821773126514865 < x < 0.030409209466777495

   1. Initial program 62.8

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

   2. Initial simplification62.8

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

   3. Taylor expanded around 0 0.0

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

   4. 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 simplification0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.026821773126514865 \lor \neg \left(x \le 0.030409209466777495\right):\\ \;\;\;\;\sqrt[3]{\frac{1}{\frac{x - \tan x}{x - \sin x}}} \cdot \left(\sqrt[3]{\frac{1}{\frac{x - \tan x}{x - \sin x}}} \cdot \sqrt[3]{\frac{1}{\frac{x - \tan x}{x - \sin x}}}\right)\\ \mathbf{else}:\\ \;\;\;\;(\left(\frac{9}{40} \cdot x\right) \cdot x + \left((\frac{-27}{2800} \cdot \left({x}^{4}\right) + \frac{-1}{2})_*\right))_*\\ \end{array}\]

Runtime

Time bar (total: 30.5s)Debug logProfile

herbie shell --seed 2018352 +o rules:numerics
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))