Average Error: 15.3 → 0.0
Time: 7.0s
Precision: binary64
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.033797675160902707 \lor \neg \left(x \le 0.0306398698667442911\right):\\ \;\;\;\;\frac{1}{\frac{\frac{x}{x - \sin x} \cdot \frac{x}{x - \sin x} - \frac{\tan x}{x - \sin x} \cdot \frac{\tan x}{x - \sin x}}{\frac{x}{x - \sin x} + \frac{1}{\sqrt[3]{x - \sin x} \cdot \sqrt[3]{x - \sin x}} \cdot \frac{\tan x}{\sqrt[3]{x - \sin x}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{9}{40} \cdot {x}^{2} - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)\\ \end{array}\]
\frac{x - \sin x}{x - \tan x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.033797675160902707 \lor \neg \left(x \le 0.0306398698667442911\right):\\
\;\;\;\;\frac{1}{\frac{\frac{x}{x - \sin x} \cdot \frac{x}{x - \sin x} - \frac{\tan x}{x - \sin x} \cdot \frac{\tan x}{x - \sin x}}{\frac{x}{x - \sin x} + \frac{1}{\sqrt[3]{x - \sin x} \cdot \sqrt[3]{x - \sin x}} \cdot \frac{\tan x}{\sqrt[3]{x - \sin 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 <= -0.03379767516090271) || !(x <= 0.03063986986674429))) {
		VAR = ((double) (1.0 / ((double) (((double) (((double) (((double) (x / ((double) (x - ((double) sin(x)))))) * ((double) (x / ((double) (x - ((double) sin(x)))))))) - ((double) (((double) (((double) tan(x)) / ((double) (x - ((double) sin(x)))))) * ((double) (((double) tan(x)) / ((double) (x - ((double) sin(x)))))))))) / ((double) (((double) (x / ((double) (x - ((double) sin(x)))))) + ((double) (((double) (1.0 / ((double) (((double) cbrt(((double) (x - ((double) sin(x)))))) * ((double) cbrt(((double) (x - ((double) sin(x)))))))))) * ((double) (((double) tan(x)) / ((double) cbrt(((double) (x - ((double) sin(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 < -0.033797675160902707 or 0.0306398698667442911 < x

    1. Initial program 0.0

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

    3. Applied clear-num0.0

      \[\leadsto \color{blue}{\frac{1}{\frac{x - \tan x}{x - \sin x}}}\]
    4. Using strategy rm

    5. Applied div-sub0.0

      \[\leadsto \frac{1}{\color{blue}{\frac{x}{x - \sin x} - \frac{\tan x}{x - \sin x}}}\]
    6. Using strategy rm

    7. Applied flip--0.0

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{x}{x - \sin x} \cdot \frac{x}{x - \sin x} - \frac{\tan x}{x - \sin x} \cdot \frac{\tan x}{x - \sin x}}{\frac{x}{x - \sin x} + \frac{\tan x}{x - \sin x}}}}\]
    8. Using strategy rm

    9. Applied add-cube-cbrt0.0

      \[\leadsto \frac{1}{\frac{\frac{x}{x - \sin x} \cdot \frac{x}{x - \sin x} - \frac{\tan x}{x - \sin x} \cdot \frac{\tan x}{x - \sin x}}{\frac{x}{x - \sin x} + \frac{\tan x}{\color{blue}{\left(\sqrt[3]{x - \sin x} \cdot \sqrt[3]{x - \sin x}\right) \cdot \sqrt[3]{x - \sin x}}}}}\]

    10. Applied *-un-lft-identity0.0

      \[\leadsto \frac{1}{\frac{\frac{x}{x - \sin x} \cdot \frac{x}{x - \sin x} - \frac{\tan x}{x - \sin x} \cdot \frac{\tan x}{x - \sin x}}{\frac{x}{x - \sin x} + \frac{\color{blue}{1 \cdot \tan x}}{\left(\sqrt[3]{x - \sin x} \cdot \sqrt[3]{x - \sin x}\right) \cdot \sqrt[3]{x - \sin x}}}}\]

    11. Applied times-frac0.0

      \[\leadsto \frac{1}{\frac{\frac{x}{x - \sin x} \cdot \frac{x}{x - \sin x} - \frac{\tan x}{x - \sin x} \cdot \frac{\tan x}{x - \sin x}}{\frac{x}{x - \sin x} + \color{blue}{\frac{1}{\sqrt[3]{x - \sin x} \cdot \sqrt[3]{x - \sin x}} \cdot \frac{\tan x}{\sqrt[3]{x - \sin x}}}}}\]

    if -0.033797675160902707 < x < 0.0306398698667442911

    1. Initial program 63.2

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

    2. Taylor expanded around 0 0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.033797675160902707 \lor \neg \left(x \le 0.0306398698667442911\right):\\ \;\;\;\;\frac{1}{\frac{\frac{x}{x - \sin x} \cdot \frac{x}{x - \sin x} - \frac{\tan x}{x - \sin x} \cdot \frac{\tan x}{x - \sin x}}{\frac{x}{x - \sin x} + \frac{1}{\sqrt[3]{x - \sin x} \cdot \sqrt[3]{x - \sin x}} \cdot \frac{\tan x}{\sqrt[3]{x - \sin x}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{9}{40} \cdot {x}^{2} - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020156 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))