Average Error: 31.7 → 0.0
Time: 10.2s
Precision: 64
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.025053611726062623:\\ \;\;\;\;\frac{\frac{\left(-{\left(\frac{\sin x}{x - \tan x}\right)}^{3}\right) \cdot \frac{\sin x}{x - \tan x} + {\left(\frac{x}{x - \tan x}\right)}^{3} \cdot \frac{x}{x - \tan x}}{\frac{x}{x - \tan x} \cdot \frac{x}{x - \tan x} + \frac{\sin x}{x - \tan x} \cdot \frac{\sin x}{x - \tan x}}}{\frac{x}{x - \tan x} + \frac{\sin x}{x - \tan x}}\\ \mathbf{elif}\;x \le 0.0271077625482928665:\\ \;\;\;\;\frac{9}{40} \cdot {x}^{2} - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\\ \end{array}\]
\frac{x - \sin x}{x - \tan x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.025053611726062623:\\
\;\;\;\;\frac{\frac{\left(-{\left(\frac{\sin x}{x - \tan x}\right)}^{3}\right) \cdot \frac{\sin x}{x - \tan x} + {\left(\frac{x}{x - \tan x}\right)}^{3} \cdot \frac{x}{x - \tan x}}{\frac{x}{x - \tan x} \cdot \frac{x}{x - \tan x} + \frac{\sin x}{x - \tan x} \cdot \frac{\sin x}{x - \tan x}}}{\frac{x}{x - \tan x} + \frac{\sin x}{x - \tan x}}\\

\mathbf{elif}\;x \le 0.0271077625482928665:\\
\;\;\;\;\frac{9}{40} \cdot {x}^{2} - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\\

\end{array}
double code(double x) {
	return ((x - sin(x)) / (x - tan(x)));
}
↓
double code(double x) {
	double temp;
	if ((x <= -0.025053611726062623)) {
		temp = ((((-pow((sin(x) / (x - tan(x))), 3.0) * (sin(x) / (x - tan(x)))) + (pow((x / (x - tan(x))), 3.0) * (x / (x - tan(x))))) / (((x / (x - tan(x))) * (x / (x - tan(x)))) + ((sin(x) / (x - tan(x))) * (sin(x) / (x - tan(x)))))) / ((x / (x - tan(x))) + (sin(x) / (x - tan(x)))));
	} else {
		double temp_1;
		if ((x <= 0.027107762548292866)) {
			temp_1 = ((0.225 * pow(x, 2.0)) - ((0.009642857142857142 * pow(x, 4.0)) + 0.5));
		} else {
			temp_1 = ((x / (x - tan(x))) - (sin(x) / (x - tan(x))));
		}
		temp = temp_1;
	}
	return temp;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if x < -0.025053611726062623

    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 flip--0.1

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

    7. Applied flip--0.1

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

    8. Simplified0.1

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

    if -0.025053611726062623 < x < 0.027107762548292866

    1. Initial program 63.2

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

    2. Taylor expanded around 0 0.0

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

    if 0.027107762548292866 < x

    1. Initial program 0.0

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

    3. Applied div-sub0.0

      \[\leadsto \color{blue}{\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.025053611726062623:\\ \;\;\;\;\frac{\frac{\left(-{\left(\frac{\sin x}{x - \tan x}\right)}^{3}\right) \cdot \frac{\sin x}{x - \tan x} + {\left(\frac{x}{x - \tan x}\right)}^{3} \cdot \frac{x}{x - \tan x}}{\frac{x}{x - \tan x} \cdot \frac{x}{x - \tan x} + \frac{\sin x}{x - \tan x} \cdot \frac{\sin x}{x - \tan x}}}{\frac{x}{x - \tan x} + \frac{\sin x}{x - \tan x}}\\ \mathbf{elif}\;x \le 0.0271077625482928665:\\ \;\;\;\;\frac{9}{40} \cdot {x}^{2} - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020058 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))