Average Error: 30.4 → 0.0
Time: 2.0m
Precision: 64
Internal Precision: 128
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.06333461761244252:\\ \;\;\;\;\frac{x}{x - \tan x} - \log \left(e^{\frac{\sin x}{x - \tan x}}\right)\\ \mathbf{elif}\;x \le 0.06506667365090828:\\ \;\;\;\;\left(\frac{9}{40} \cdot \left(x \cdot x\right) - \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{27}{2800}\right) - \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x - \tan x} - \log \left(e^{\frac{\sin x}{x - \tan x}}\right)\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -0.06333461761244252 or 0.06506667365090828 < x

    1. Initial program 0.0

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

    3. Applied add-log-exp0.0

      \[\leadsto \color{blue}{\log \left(e^{\frac{x - \sin x}{x - \tan x}}\right)}\]
    4. Using strategy rm

    5. Applied div-sub0.1

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

    6. Applied exp-diff0.1

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

    7. Applied log-div0.1

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

    8. Simplified0.1

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

    if -0.06333461761244252 < x < 0.06506667365090828

    1. Initial program 62.6

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

    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

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

    5. Applied associate--r+0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.06333461761244252:\\ \;\;\;\;\frac{x}{x - \tan x} - \log \left(e^{\frac{\sin x}{x - \tan x}}\right)\\ \mathbf{elif}\;x \le 0.06506667365090828:\\ \;\;\;\;\left(\frac{9}{40} \cdot \left(x \cdot x\right) - \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{27}{2800}\right) - \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x - \tan x} - \log \left(e^{\frac{\sin x}{x - \tan x}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019088 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))