Average Error: 31.0 → 0.1
Time: 11.6s
Precision: binary64
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.55226078239205623 \lor \neg \left(x \le 1.593823631543573\right):\\ \;\;\;\;\sqrt{\frac{x - \sin x}{x - \tan x}} \cdot \sqrt{\frac{x - \sin x}{x - \tan x}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{9}{40} \cdot {x}^{2} - \frac{27}{2800} \cdot {x}^{4}\right) - \frac{1}{2}\\ \end{array}\]
\frac{x - \sin x}{x - \tan x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.55226078239205623 \lor \neg \left(x \le 1.593823631543573\right):\\
\;\;\;\;\sqrt{\frac{x - \sin x}{x - \tan x}} \cdot \sqrt{\frac{x - \sin x}{x - \tan x}}\\

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

\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 <= -1.5522607823920562) || !(x <= 1.593823631543573))) {
		VAR = ((double) (((double) sqrt(((double) (((double) (x - ((double) sin(x)))) / ((double) (x - ((double) tan(x)))))))) * ((double) sqrt(((double) (((double) (x - ((double) sin(x)))) / ((double) (x - ((double) tan(x))))))))));
	} else {
		VAR = ((double) (((double) (((double) (0.225 * ((double) pow(x, 2.0)))) - ((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 < -1.55226078239205623 or 1.593823631543573 < x

    1. Initial program 0.0

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

    3. Applied add-sqr-sqrt0.0

      \[\leadsto \color{blue}{\sqrt{\frac{x - \sin x}{x - \tan x}} \cdot \sqrt{\frac{x - \sin x}{x - \tan x}}}\]

    if -1.55226078239205623 < x < 1.593823631543573

    1. Initial program 62.8

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

    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. Using strategy rm

    4. Applied associate--r+0.2

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.55226078239205623 \lor \neg \left(x \le 1.593823631543573\right):\\ \;\;\;\;\sqrt{\frac{x - \sin x}{x - \tan x}} \cdot \sqrt{\frac{x - \sin x}{x - \tan x}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{9}{40} \cdot {x}^{2} - \frac{27}{2800} \cdot {x}^{4}\right) - \frac{1}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020171 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))