Average Error: 31.3 → 0.1
Time: 9.0s
Precision: binary64
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.030804876869537152 \lor \neg \left(x \le 0.026590185665342832\right):\\ \;\;\;\;\left(\sqrt[3]{\frac{x - \sin x}{x - \tan x}} \cdot \sqrt[3]{\frac{x - \sin x}{x - \tan x}}\right) \cdot \sqrt[3]{\frac{x - \sin x}{x - \tan x}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{9}{40} \cdot {x}^{2} - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)}\right)\\ \end{array}\]
\frac{x - \sin x}{x - \tan x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.030804876869537152 \lor \neg \left(x \le 0.026590185665342832\right):\\
\;\;\;\;\left(\sqrt[3]{\frac{x - \sin x}{x - \tan x}} \cdot \sqrt[3]{\frac{x - \sin x}{x - \tan x}}\right) \cdot \sqrt[3]{\frac{x - \sin x}{x - \tan x}}\\

\mathbf{else}:\\
\;\;\;\;\log \left(e^{\frac{9}{40} \cdot {x}^{2} - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)}\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 <= -0.030804876869537152) || !(x <= 0.02659018566534283))) {
		VAR = ((double) (((double) (((double) cbrt(((double) (((double) (x - ((double) sin(x)))) / ((double) (x - ((double) tan(x)))))))) * ((double) cbrt(((double) (((double) (x - ((double) sin(x)))) / ((double) (x - ((double) tan(x)))))))))) * ((double) cbrt(((double) (((double) (x - ((double) sin(x)))) / ((double) (x - ((double) tan(x))))))))));
	} else {
		VAR = ((double) log(((double) exp(((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 < -0.030804876869537152 or 0.026590185665342832 < x

    1. Initial program 0.0

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

    3. Applied add-cube-cbrt0.1

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

    if -0.030804876869537152 < x < 0.026590185665342832

    1. Initial program 63.3

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

    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. Using strategy rm

    4. Applied add-log-exp0.0

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

    5. Applied add-log-exp0.0

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

    6. Applied sum-log0.0

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

    7. Applied add-log-exp0.0

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

    8. Applied diff-log0.0

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

    9. Simplified0.0

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.030804876869537152 \lor \neg \left(x \le 0.026590185665342832\right):\\ \;\;\;\;\left(\sqrt[3]{\frac{x - \sin x}{x - \tan x}} \cdot \sqrt[3]{\frac{x - \sin x}{x - \tan x}}\right) \cdot \sqrt[3]{\frac{x - \sin x}{x - \tan x}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{9}{40} \cdot {x}^{2} - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020149 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))