Average Error: 31.4 → 0.0
Time: 33.9s
Precision: 64
Internal Precision: 2368
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.03171362506925326:\\ \;\;\;\;\sqrt[3]{\frac{x - \sin x}{x - \tan x} \cdot \left(\frac{x - \sin x}{x - \tan x} \cdot \frac{x - \sin x}{x - \tan x}\right)}\\ \mathbf{elif}\;x \le 0.027271857488427573:\\ \;\;\;\;(\left(\frac{9}{40} \cdot x\right) \cdot x + \left((\frac{-27}{2800} \cdot \left({x}^{4}\right) + \frac{-1}{2})_*\right))_*\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(\frac{x - \sin x}{x - \tan x} \cdot \frac{x - \sin x}{x - \tan x}\right) \cdot \frac{1}{\frac{x - \tan x}{x - \sin x}}}\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 3 regimes

  2. if x < -0.03171362506925326

    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 add-cbrt-cube0.1

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

    if -0.03171362506925326 < x < 0.027271857488427573

    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

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

    if 0.027271857488427573 < 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 add-cbrt-cube0.1

      \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{x - \sin x}{x - \tan x} \cdot \frac{x - \sin x}{x - \tan x}\right) \cdot \frac{x - \sin x}{x - \tan x}}}\]
    5. Using strategy rm

    6. Applied clear-num0.1

      \[\leadsto \sqrt[3]{\left(\frac{x - \sin x}{x - \tan x} \cdot \frac{x - \sin x}{x - \tan x}\right) \cdot \color{blue}{\frac{1}{\frac{x - \tan x}{x - \sin x}}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.03171362506925326:\\ \;\;\;\;\sqrt[3]{\frac{x - \sin x}{x - \tan x} \cdot \left(\frac{x - \sin x}{x - \tan x} \cdot \frac{x - \sin x}{x - \tan x}\right)}\\ \mathbf{elif}\;x \le 0.027271857488427573:\\ \;\;\;\;(\left(\frac{9}{40} \cdot x\right) \cdot x + \left((\frac{-27}{2800} \cdot \left({x}^{4}\right) + \frac{-1}{2})_*\right))_*\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(\frac{x - \sin x}{x - \tan x} \cdot \frac{x - \sin x}{x - \tan x}\right) \cdot \frac{1}{\frac{x - \tan x}{x - \sin x}}}\\ \end{array}\]

Runtime

Time bar (total: 33.9s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes31.70.00.031.6100%
herbie shell --seed 2018274 +o rules:numerics
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))