Average Error: 31.4 → 0.0
Time: 10.8min
Precision: binary64
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.0353632816255217072 \lor \neg \left(x \le 0.0268834472773056\right):\\ \;\;\;\;\frac{x - \sqrt[3]{{\left(\sin x\right)}^{3}}}{x - \tan x}\\ \mathbf{else}:\\ \;\;\;\;\frac{9}{40} \cdot {x}^{2} - \left(\frac{1}{2} + \frac{27}{2800} \cdot {x}^{4}\right)\\ \end{array}\]
\frac{x - \sin x}{x - \tan x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.0353632816255217072 \lor \neg \left(x \le 0.0268834472773056\right):\\
\;\;\;\;\frac{x - \sqrt[3]{{\left(\sin x\right)}^{3}}}{x - \tan x}\\

\mathbf{else}:\\
\;\;\;\;\frac{9}{40} \cdot {x}^{2} - \left(\frac{1}{2} + \frac{27}{2800} \cdot {x}^{4}\right)\\

\end{array}
double code(double x) {
	return (((double) (x - ((double) sin(x)))) / ((double) (x - ((double) tan(x)))));
}
↓
double code(double x) {
	double VAR;
	if (((x <= -0.03536328162552171) || !(x <= 0.0268834472773056))) {
		VAR = (((double) (x - ((double) cbrt(((double) pow(((double) sin(x)), 3.0)))))) / ((double) (x - ((double) tan(x)))));
	} else {
		VAR = ((double) (((double) (0.225 * ((double) pow(x, 2.0)))) - ((double) (0.5 + ((double) (0.009642857142857142 * ((double) pow(x, 4.0))))))));
	}
	return VAR;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if x < -0.0353632816255217072 or 0.0268834472773056 < x

    1. Initial program 0.0

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

    3. Applied add-cbrt-cube0.0

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

    4. Simplified0.1

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

    if -0.0353632816255217072 < x < 0.0268834472773056

    1. Initial program 63.0

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

    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. Recombined 2 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0353632816255217072 \lor \neg \left(x \le 0.0268834472773056\right):\\ \;\;\;\;\frac{x - \sqrt[3]{{\left(\sin x\right)}^{3}}}{x - \tan x}\\ \mathbf{else}:\\ \;\;\;\;\frac{9}{40} \cdot {x}^{2} - \left(\frac{1}{2} + \frac{27}{2800} \cdot {x}^{4}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020181 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))