Average Error: 30.9 → 0.0
Time: 3.9m
Precision: 64
Internal Precision: 2624
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.03243492521180331 \lor \neg \left(x \le 0.02509086206773924\right):\\ \;\;\;\;\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(x \cdot \left(x \cdot \frac{9}{40}\right)\right)}^{3} - {\left((\frac{27}{2800} \cdot \left({x}^{4}\right) + \frac{1}{2})_*\right)}^{3}}{(\left((\frac{27}{2800} \cdot \left({x}^{4}\right) + \frac{1}{2})_*\right) \cdot \left((\left(x \cdot x\right) \cdot \frac{9}{40} + \left((\frac{27}{2800} \cdot \left({x}^{4}\right) + \frac{1}{2})_*\right))_*\right) + \left(\left(x \cdot \left(x \cdot \frac{9}{40}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{9}{40}\right)\right)\right))_*}\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -0.03243492521180331 or 0.02509086206773924 < x

    1. Initial program 0.1

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

    3. Applied div-sub0.1

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

    if -0.03243492521180331 < x < 0.02509086206773924

    1. Initial program 62.8

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

    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

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

    5. Applied flip3--0.0

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

    6. Applied simplify0.0

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

  4. Applied simplify0.0

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;x \le -0.03243492521180331 \lor \neg \left(x \le 0.02509086206773924\right):\\ \;\;\;\;\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(x \cdot \left(x \cdot \frac{9}{40}\right)\right)}^{3} - {\left((\frac{27}{2800} \cdot \left({x}^{4}\right) + \frac{1}{2})_*\right)}^{3}}{(\left((\frac{27}{2800} \cdot \left({x}^{4}\right) + \frac{1}{2})_*\right) \cdot \left((\left(x \cdot x\right) \cdot \frac{9}{40} + \left((\frac{27}{2800} \cdot \left({x}^{4}\right) + \frac{1}{2})_*\right))_*\right) + \left(\left(x \cdot \left(x \cdot \frac{9}{40}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{9}{40}\right)\right)\right))_*}\\ \end{array}}\]

Runtime

Time bar (total: 3.9m)Debug logProfile

herbie shell --seed 2018208 +o rules:numerics
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))