Error

Derivation

1. Split input into 2 regimes

2. if x < -0.029982417226586153 or 0.0276643419913582 < x

   1. Initial program 0.1

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

   3. Applied div-sub0.1

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

   if -0.029982417226586153 < x < 0.0276643419913582

   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

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

   5. Applied add-log-exp0.0

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

4. Final simplification0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.029982417226586153:\\ \;\;\;\;\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\\ \mathbf{elif}\;x \le 0.0276643419913582:\\ \;\;\;\;\log \left(e^{(\left(x \cdot x\right) \cdot \frac{9}{40} + \left((\frac{-27}{2800} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) + \frac{-1}{2})_*\right))_*}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019094 +o rules:numerics
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))