Average Error: 31.2 → 0.0
Time: 1.2m
Precision: 64
Internal Precision: 128
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.028121595842453516:\\ \;\;\;\;\frac{x}{x - \tan x} + \frac{-\sin x}{x - \tan x}\\ \mathbf{elif}\;x \le 0.026925532592986075:\\ \;\;\;\;(\left({x}^{4}\right) \cdot \frac{-27}{2800} + \left((\left(\frac{9}{40} \cdot x\right) \cdot x + \frac{-1}{2})_*\right))_*\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\right) \cdot \left(\left(\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\right) \cdot \left(\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\right)\right)}\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 3 regimes

  2. if x < -0.028121595842453516

    1. Initial program 0.1

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

    3. Applied div-sub0.1

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

    5. Applied sub-neg0.1

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

    if -0.028121595842453516 < x < 0.026925532592986075

    1. Initial program 62.7

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

    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

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

    if 0.026925532592986075 < x

    1. Initial program 0.1

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

    3. Applied div-sub0.1

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

    5. Applied add-cbrt-cube0.1

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.028121595842453516:\\ \;\;\;\;\frac{x}{x - \tan x} + \frac{-\sin x}{x - \tan x}\\ \mathbf{elif}\;x \le 0.026925532592986075:\\ \;\;\;\;(\left({x}^{4}\right) \cdot \frac{-27}{2800} + \left((\left(\frac{9}{40} \cdot x\right) \cdot x + \frac{-1}{2})_*\right))_*\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\right) \cdot \left(\left(\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\right) \cdot \left(\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019053 +o rules:numerics
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))