Average Error: 31.6 → 0.0
Time: 1.6m
Precision: 64
Internal Precision: 2368
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.028222696665668904 \lor \neg \left(x \le 0.02715967922074874\right):\\ \;\;\;\;\frac{\frac{\frac{x}{x - \tan x} \cdot x - \frac{\sin x}{x - \tan x} \cdot \sin x}{x - \tan x}}{\frac{x}{x - \tan x} + \frac{\sin x}{x - \tan x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{9}{40} \cdot {x}^{2} - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)\\ \end{array}\]

Error

Bits error versus x

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Results

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Derivation

  1. Split input into 2 regimes

  2. if x < -0.028222696665668904 or 0.02715967922074874 < x

    1. Initial program 0.0

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

    3. Applied div-sub0.0

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

    5. Applied flip--0.0

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

    7. Applied associate-*r/0.0

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

    8. Applied associate-*r/0.0

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

    9. Applied sub-div0.0

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

    if -0.028222696665668904 < x < 0.02715967922074874

    1. Initial program 62.9

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

    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. Recombined 2 regimes into one program.

  4. Applied simplify0.0

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;x \le -0.028222696665668904 \lor \neg \left(x \le 0.02715967922074874\right):\\ \;\;\;\;\frac{\frac{\frac{x}{x - \tan x} \cdot x - \frac{\sin x}{x - \tan x} \cdot \sin x}{x - \tan x}}{\frac{x}{x - \tan x} + \frac{\sin x}{x - \tan x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{9}{40} \cdot {x}^{2} - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)\\ \end{array}}\]

Runtime

Time bar (total: 1.6m)Debug logProfile

herbie shell --seed 2018170 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))