Average Error: 31.4 → 0.0
Time: 10.9s
Precision: binary64
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.028030499297402853 \lor \neg \left(x \le 0.0216525873793473704\right):\\ \;\;\;\;\frac{{\left(\frac{x}{x - \tan x}\right)}^{3} - {\left(\frac{\sin x}{x - \tan x}\right)}^{3}}{\frac{\sin x}{x - \tan x} \cdot \left(\frac{\sin x}{x - \tan x} + \frac{x}{x - \tan x}\right) + \frac{x}{x - \tan x} \cdot \frac{x}{x - \tan x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{9}{40} \cdot {x}^{2} - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)\\ \end{array}\]
\frac{x - \sin x}{x - \tan x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.028030499297402853 \lor \neg \left(x \le 0.0216525873793473704\right):\\
\;\;\;\;\frac{{\left(\frac{x}{x - \tan x}\right)}^{3} - {\left(\frac{\sin x}{x - \tan x}\right)}^{3}}{\frac{\sin x}{x - \tan x} \cdot \left(\frac{\sin x}{x - \tan x} + \frac{x}{x - \tan x}\right) + \frac{x}{x - \tan x} \cdot \frac{x}{x - \tan x}}\\

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

\end{array}
double code(double x) {
	return ((double) (((double) (x - ((double) sin(x)))) / ((double) (x - ((double) tan(x))))));
}
↓
double code(double x) {
	double VAR;
	if (((x <= -0.028030499297402853) || !(x <= 0.02165258737934737))) {
		VAR = ((double) (((double) (((double) pow(((double) (x / ((double) (x - ((double) tan(x)))))), 3.0)) - ((double) pow(((double) (((double) sin(x)) / ((double) (x - ((double) tan(x)))))), 3.0)))) / ((double) (((double) (((double) (((double) sin(x)) / ((double) (x - ((double) tan(x)))))) * ((double) (((double) (((double) sin(x)) / ((double) (x - ((double) tan(x)))))) + ((double) (x / ((double) (x - ((double) tan(x)))))))))) + ((double) (((double) (x / ((double) (x - ((double) tan(x)))))) * ((double) (x / ((double) (x - ((double) tan(x))))))))))));
	} else {
		VAR = ((double) (((double) (0.225 * ((double) pow(x, 2.0)))) - ((double) (((double) (0.009642857142857142 * ((double) pow(x, 4.0)))) + 0.5))));
	}
	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 < -0.028030499297402853 or 0.02165258737934737 < x

    1. Initial program 0.1

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

    3. Applied div-sub0.1

      \[\leadsto \color{blue}{\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}}\]
    4. Using strategy rm

    5. Applied flip3--0.1

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

    6. Simplified0.1

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

    if -0.028030499297402853 < x < 0.02165258737934737

    1. Initial program 63.1

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

    2. Taylor expanded around 0 0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.028030499297402853 \lor \neg \left(x \le 0.0216525873793473704\right):\\ \;\;\;\;\frac{{\left(\frac{x}{x - \tan x}\right)}^{3} - {\left(\frac{\sin x}{x - \tan x}\right)}^{3}}{\frac{\sin x}{x - \tan x} \cdot \left(\frac{\sin x}{x - \tan x} + \frac{x}{x - \tan x}\right) + \frac{x}{x - \tan x} \cdot \frac{x}{x - \tan x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{9}{40} \cdot {x}^{2} - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020148 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))