Average Error: 31.2 → 0.1
Time: 2.0m
Precision: 64
Internal Precision: 2368
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \cdot \left(\frac{9}{40} \cdot x\right) - (\frac{27}{2800} \cdot \left({x}^{4}\right) + \frac{1}{2})_* \le -0.5125171501916893 \lor \neg \left(x \cdot \left(\frac{9}{40} \cdot x\right) - (\frac{27}{2800} \cdot \left({x}^{4}\right) + \frac{1}{2})_* \le -0.4989573931249496\right):\\ \;\;\;\;(\left(\sqrt[3]{\frac{x}{x - \tan x}} \cdot \sqrt[3]{\frac{x}{x - \tan x}}\right) \cdot \left(\sqrt[3]{\frac{x}{x - \tan x}}\right) + \left(\frac{-\sin x}{x - \tan x}\right))_*\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{9}{40} \cdot x\right) - (\frac{27}{2800} \cdot \left({x}^{4}\right) + \frac{1}{2})_*\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if (- (* x (* x 9/40)) (fma 27/2800 (pow x 4) 1/2)) < -0.5125171501916893 or -0.4989573931249496 < (- (* x (* x 9/40)) (fma 27/2800 (pow x 4) 1/2))

    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 add-cube-cbrt0.1

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

    6. Applied fma-neg0.1

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

    if -0.5125171501916893 < (- (* x (* x 9/40)) (fma 27/2800 (pow x 4) 1/2)) < -0.4989573931249496

    1. Initial program 62.6

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

    2. Taylor expanded around 0 0.0

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

    3. Applied simplify0.0

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \frac{9}{40}\right) - (\frac{27}{2800} \cdot \left({x}^{4}\right) + \frac{1}{2})_*}\]
  3. Recombined 2 regimes into one program.

  4. Applied simplify0.1

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;x \cdot \left(\frac{9}{40} \cdot x\right) - (\frac{27}{2800} \cdot \left({x}^{4}\right) + \frac{1}{2})_* \le -0.5125171501916893 \lor \neg \left(x \cdot \left(\frac{9}{40} \cdot x\right) - (\frac{27}{2800} \cdot \left({x}^{4}\right) + \frac{1}{2})_* \le -0.4989573931249496\right):\\ \;\;\;\;(\left(\sqrt[3]{\frac{x}{x - \tan x}} \cdot \sqrt[3]{\frac{x}{x - \tan x}}\right) \cdot \left(\sqrt[3]{\frac{x}{x - \tan x}}\right) + \left(\frac{-\sin x}{x - \tan x}\right))_*\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{9}{40} \cdot x\right) - (\frac{27}{2800} \cdot \left({x}^{4}\right) + \frac{1}{2})_*\\ \end{array}}\]

Runtime

Time bar (total: 2.0m)Debug logProfile

herbie shell --seed 2018193 +o rules:numerics
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))