Average Error: 30.7 → 0.0
Time: 40.8s
Precision: 64
Internal Precision: 2368
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.028308868741759505:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \mathbf{elif}\;x \le 0.028204893689164384:\\ \;\;\;\;\left(\left(x \cdot \frac{9}{40}\right) \cdot x - \frac{27}{2800} \cdot {x}^{4}\right) - \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(\frac{x - \sin x}{x - \tan x} \cdot \frac{x - \sin x}{x - \tan x}\right) \cdot \frac{x - \sin x}{x - \tan x}}\\ \end{array}\]

Error

Bits error versus x

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Results

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Derivation

  1. Split input into 3 regimes

  2. if x < -0.028308868741759505

    1. Initial program 0.0

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

    2. Taylor expanded around inf 0.0

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

    if -0.028308868741759505 < x < 0.028204893689164384

    1. Initial program 62.9

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

    2. Taylor expanded around inf 62.9

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

    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

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

    6. Applied fma-udef0.0

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

    7. Applied associate--r+0.0

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

    if 0.028204893689164384 < x

    1. Initial program 0.1

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

    2. Taylor expanded around inf 0.1

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

    4. Applied add-cbrt-cube0.1

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.028308868741759505:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \mathbf{elif}\;x \le 0.028204893689164384:\\ \;\;\;\;\left(\left(x \cdot \frac{9}{40}\right) \cdot x - \frac{27}{2800} \cdot {x}^{4}\right) - \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(\frac{x - \sin x}{x - \tan x} \cdot \frac{x - \sin x}{x - \tan x}\right) \cdot \frac{x - \sin x}{x - \tan x}}\\ \end{array}\]

Runtime

Time bar (total: 40.8s)Debug logProfile

herbie shell --seed 2018227 +o rules:numerics
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))