Average Error: 32.2 → 0.3
Time: 10.8s
Precision: 64
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2.4709346097908487 \lor \neg \left(x \le 2.4056784657088475\right):\\ \;\;\;\;\left(\frac{\sin x}{x \cdot \cos x} + \left(\frac{{\left(\sin x\right)}^{2}}{{x}^{2} \cdot {\left(\cos x\right)}^{2}} + 1\right)\right) - \left(\frac{\sin x}{x} + \frac{{\left(\sin x\right)}^{2}}{{x}^{2} \cdot \cos x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{9}{40} \cdot {x}^{2} - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)\\ \end{array}\]
\frac{x - \sin x}{x - \tan x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -2.4709346097908487 \lor \neg \left(x \le 2.4056784657088475\right):\\
\;\;\;\;\left(\frac{\sin x}{x \cdot \cos x} + \left(\frac{{\left(\sin x\right)}^{2}}{{x}^{2} \cdot {\left(\cos x\right)}^{2}} + 1\right)\right) - \left(\frac{\sin x}{x} + \frac{{\left(\sin x\right)}^{2}}{{x}^{2} \cdot \cos x}\right)\\

\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 <= -2.4709346097908487) || !(x <= 2.4056784657088475))) {
		VAR = ((double) (((double) (((double) (((double) sin(x)) / ((double) (x * ((double) cos(x)))))) + ((double) (((double) (((double) pow(((double) sin(x)), 2.0)) / ((double) (((double) pow(x, 2.0)) * ((double) pow(((double) cos(x)), 2.0)))))) + 1.0)))) - ((double) (((double) (((double) sin(x)) / x)) + ((double) (((double) pow(((double) sin(x)), 2.0)) / ((double) (((double) pow(x, 2.0)) * ((double) cos(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 < -2.4709346097908487 or 2.4056784657088475 < x

    1. Initial program 0.0

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

    2. Taylor expanded around inf 0.4

      \[\leadsto \color{blue}{\left(\frac{\sin x}{x \cdot \cos x} + \left(\frac{{\left(\sin x\right)}^{2}}{{x}^{2} \cdot {\left(\cos x\right)}^{2}} + 1\right)\right) - \left(\frac{\sin x}{x} + \frac{{\left(\sin x\right)}^{2}}{{x}^{2} \cdot \cos x}\right)}\]

    if -2.4709346097908487 < x < 2.4056784657088475

    1. Initial program 62.7

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

    2. Taylor expanded around 0 0.2

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.4709346097908487 \lor \neg \left(x \le 2.4056784657088475\right):\\ \;\;\;\;\left(\frac{\sin x}{x \cdot \cos x} + \left(\frac{{\left(\sin x\right)}^{2}}{{x}^{2} \cdot {\left(\cos x\right)}^{2}} + 1\right)\right) - \left(\frac{\sin x}{x} + \frac{{\left(\sin x\right)}^{2}}{{x}^{2} \cdot \cos x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{9}{40} \cdot {x}^{2} - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020113 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))