Average Error: 31.1 → 0.0
Time: 2.0m
Precision: 64
Internal Precision: 2432
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.03021091695231464:\\ \;\;\;\;\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\\ \mathbf{if}\;x \le 0.03040127665186386:\\ \;\;\;\;x \cdot \left(x \cdot \frac{9}{40}\right) - (\frac{27}{2800} \cdot \left({x}^{4}\right) + \frac{1}{2})_*\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -0.03021091695231464 or 0.03040127665186386 < 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 -0.03021091695231464 < x < 0.03040127665186386

    1. Initial program 62.8

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

    2. Taylor expanded around 0 0.0

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

    3. Applied simplify0.0

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

Runtime

Time bar (total: 2.0m)Debug logProfile

herbie shell --seed '#(1070833653 108281690 3330367898 3632331308 3494323072 43156186)' +o rules:numerics
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))