Average Error: 31.3 → 0.0
Time: 46.6s
Precision: 64
Internal Precision: 2624
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.03222411668912717 \lor \neg \left(x \le 0.02881131157430173\right):\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \frac{9}{40}\right) \cdot x - (\left({x}^{4}\right) \cdot \frac{27}{2800} + \frac{1}{2})_*\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -0.03222411668912717 or 0.02881131157430173 < x

    1. Initial program 0.0

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

    if -0.03222411668912717 < x < 0.02881131157430173

    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

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.03222411668912717 \lor \neg \left(x \le 0.02881131157430173\right):\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \frac{9}{40}\right) \cdot x - (\left({x}^{4}\right) \cdot \frac{27}{2800} + \frac{1}{2})_*\\ \end{array}\]

Runtime

Time bar (total: 46.6s)Debug logProfile

herbie shell --seed 2018221 +o rules:numerics
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))