Average Error: 31.5 → 0.1
Time: 37.3s
Precision: 64
Internal precision: 2432
\[\frac{x - \sin x}{x - \tan x}\]
⬇
\[\begin{array}{l}
\mathbf{if}\;x \le -9.533351186782151 \cdot 10^{-07}:\\
\;\;\;\;\sqrt[3]{{\left(\frac{x - \sin x}{x - \tan x}\right)}^3}\\
\mathbf{if}\;x \le 5126479.956417156:\\
\;\;\;\;\frac{9}{40} \cdot {x}^2 - \left(\frac{1}{2} + \frac{27}{2800} \cdot {x}^{4}\right)\\
\mathbf{else}:\\
\;\;\;\;\frac{x - \sin x}{x - \tan x}\\
\end{array}\]
Derivation
- Split input into 3 regimes.
-
if x < -9.533351186782151e-07
Initial program 0.3
\[\frac{x - \sin x}{x - \tan x}\]
- Using strategy
rm
Applied add-cbrt-cube 41.6
\[\leadsto \frac{x - \sin x}{\color{blue}{\sqrt[3]{{\left(x - \tan x\right)}^3}}}\]
Applied add-cbrt-cube 41.5
\[\leadsto \frac{\color{blue}{\sqrt[3]{{\left(x - \sin x\right)}^3}}}{\sqrt[3]{{\left(x - \tan x\right)}^3}}\]
Applied cbrt-undiv 41.5
\[\leadsto \color{blue}{\sqrt[3]{\frac{{\left(x - \sin x\right)}^3}{{\left(x - \tan x\right)}^3}}}\]
Applied simplify 0.3
\[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{x - \sin x}{x - \tan x}\right)}^3}}\]
if -9.533351186782151e-07 < x < 5126479.956417156
Initial program 63.0
\[\frac{x - \sin x}{x - \tan x}\]
Applied taylor 0.0
\[\leadsto \frac{9}{40} \cdot {x}^2 - \left(\frac{1}{2} + \frac{27}{2800} \cdot {x}^{4}\right)\]
Taylor expanded around 0 0.0
\[\leadsto \color{blue}{\frac{9}{40} \cdot {x}^2 - \left(\frac{1}{2} + \frac{27}{2800} \cdot {x}^{4}\right)}\]
if 5126479.956417156 < x
Initial program 0.0
\[\frac{x - \sin x}{x - \tan x}\]
- Recombined 3 regimes into one program.
- Removed slow pow expressions
Runtime
Please include this information when filing a bug report:
herbie shell --seed '#(3283856077 3183919125 3399458751 396155847 3688194147 1862413033)'
(FPCore (x)
:name "sintan (problem 3.4.5)"
(/ (- x (sin x)) (- x (tan x))))
