Average Error: 31.0 → 0.0
Time: 1.7m
Precision: 64
Internal Precision: 128
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.031478185219520595 \lor \neg \left(x \le 0.0295278792098092\right):\\ \;\;\;\;\frac{{\left(\frac{x}{x - \tan x}\right)}^{3} - {\left(\frac{\sin x}{x - \tan x}\right)}^{3}}{\left(\frac{\sin x}{x - \tan x} \cdot \frac{\sin x}{x - \tan x} + \frac{\sin x}{x - \tan x} \cdot \frac{x}{x - \tan x}\right) + \frac{x}{x - \tan x} \cdot \frac{x}{x - \tan x}}\\ \mathbf{else}:\\ \;\;\;\;(\left(x \cdot \frac{9}{40}\right) \cdot x + \left((\frac{-27}{2800} \cdot \left({x}^{4}\right) + \frac{-1}{2})_*\right))_*\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -0.031478185219520595 or 0.0295278792098092 < x

    1. Initial program 0.0

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

    2. Initial simplification0.0

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

    4. Applied div-sub0.1

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

    6. Applied flip3--0.1

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

    if -0.031478185219520595 < x < 0.0295278792098092

    1. Initial program 62.6

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

    2. Initial simplification62.6

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

    4. Applied div-sub62.6

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

    5. Taylor expanded around 0 0.0

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

    6. 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 simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.031478185219520595 \lor \neg \left(x \le 0.0295278792098092\right):\\ \;\;\;\;\frac{{\left(\frac{x}{x - \tan x}\right)}^{3} - {\left(\frac{\sin x}{x - \tan x}\right)}^{3}}{\left(\frac{\sin x}{x - \tan x} \cdot \frac{\sin x}{x - \tan x} + \frac{\sin x}{x - \tan x} \cdot \frac{x}{x - \tan x}\right) + \frac{x}{x - \tan x} \cdot \frac{x}{x - \tan x}}\\ \mathbf{else}:\\ \;\;\;\;(\left(x \cdot \frac{9}{40}\right) \cdot x + \left((\frac{-27}{2800} \cdot \left({x}^{4}\right) + \frac{-1}{2})_*\right))_*\\ \end{array}\]

Runtime

Time bar (total: 1.7m)Debug logProfile

herbie shell --seed 2018296 +o rules:numerics
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))