Average Error: 31.6 → 0.1
Time: 20.0s
Precision: 64
Internal precision: 2432
\[\frac{x - \sin x}{x - \tan x}\]
⬇
\[\begin{array}{l}
\mathbf{if}\;x \le -2.3416378887887987 \cdot 10^{-05}:\\
\;\;\;\;\sqrt[3]{{\left(\frac{x - \sin x}{x - \tan x}\right)}^3}\\
\mathbf{if}\;x \le 1843.611849545597:\\
\;\;\;\;\frac{9}{40} \cdot {x}^2 - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)\\
\mathbf{else}:\\
\;\;\;\;\frac{1}{\frac{x - \tan x}{x - \sin x}}\\
\end{array}\]
Derivation
- Split input into 3 regimes.
-
if x < -2.3416378887887987e-05
Initial program 0.3
\[\frac{x - \sin x}{x - \tan x}\]
- Using strategy
rm
Applied add-cbrt-cube 40.4
\[\leadsto \frac{x - \sin x}{\color{blue}{\sqrt[3]{{\left(x - \tan x\right)}^3}}}\]
Applied add-cbrt-cube 40.3
\[\leadsto \frac{\color{blue}{\sqrt[3]{{\left(x - \sin x\right)}^3}}}{\sqrt[3]{{\left(x - \tan x\right)}^3}}\]
Applied cbrt-undiv 40.3
\[\leadsto \color{blue}{\sqrt[3]{\frac{{\left(x - \sin x\right)}^3}{{\left(x - \tan x\right)}^3}}}\]
Applied simplify 0.3
\[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{x - \sin x}{x - \tan x}\right)}^3}}\]
if -2.3416378887887987e-05 < x < 1843.611849545597
Initial program 62.8
\[\frac{x - \sin x}{x - \tan x}\]
Applied taylor 0.0
\[\leadsto \frac{9}{40} \cdot {x}^2 - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)\]
Taylor expanded around 0 0.0
\[\leadsto \color{blue}{\frac{9}{40} \cdot {x}^2 - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)}\]
if 1843.611849545597 < x
Initial program 0.0
\[\frac{x - \sin x}{x - \tan x}\]
- Using strategy
rm
Applied clear-num 0.0
\[\leadsto \color{blue}{\frac{1}{\frac{x - \tan x}{x - \sin x}}}\]
- Recombined 3 regimes into one program.
- Removed slow pow expressions
Runtime
Please include this information when filing a bug report:
herbie shell --seed '#(1064555532 179913862 452496668 2441903500 287849034 462453547)'
(FPCore (x)
:name "sintan (problem 3.4.5)"
(/ (- x (sin x)) (- x (tan x))))
