Average Error: 31.9 → 0.0
Time: 8.7s
Precision: binary64
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.026253755058246165 \lor \neg \left(x \le 0.0282136986304838445\right):\\ \;\;\;\;\log \left(e^{\frac{x - \sin x}{x - \tan x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} + \left({x}^{4} \cdot \frac{-27}{2800} + x \cdot \left(x \cdot \frac{9}{40}\right)\right)\\ \end{array}\]
\frac{x - \sin x}{x - \tan x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.026253755058246165 \lor \neg \left(x \le 0.0282136986304838445\right):\\
\;\;\;\;\log \left(e^{\frac{x - \sin x}{x - \tan x}}\right)\\

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

\end{array}
double code(double x) {
	return (((double) (x - ((double) sin(x)))) / ((double) (x - ((double) tan(x)))));
}
↓
double code(double x) {
	double VAR;
	if (((x <= -0.026253755058246165) || !(x <= 0.028213698630483845))) {
		VAR = ((double) log(((double) exp((((double) (x - ((double) sin(x)))) / ((double) (x - ((double) tan(x)))))))));
	} else {
		VAR = ((double) (-0.5 + ((double) (((double) (((double) pow(x, 4.0)) * -0.009642857142857142)) + ((double) (x * ((double) (x * 0.225))))))));
	}
	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 < -0.026253755058246165 or 0.0282136986304838445 < x

    1. Initial program 0.0

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

    3. Applied add-log-exp0.0

      \[\leadsto \color{blue}{\log \left(e^{\frac{x - \sin x}{x - \tan x}}\right)}\]

    if -0.026253755058246165 < x < 0.0282136986304838445

    1. Initial program 63.0

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

    3. Applied add-log-exp63.0

      \[\leadsto \color{blue}{\log \left(e^{\frac{x - \sin x}{x - \tan x}}\right)}\]

    4. Taylor expanded around 0 0.0

      \[\leadsto \log \color{blue}{\left(\frac{351}{22400} \cdot \left({x}^{4} \cdot e^{\frac{-1}{2}}\right) + \left(e^{\frac{-1}{2}} + \frac{9}{40} \cdot \left({x}^{2} \cdot e^{\frac{-1}{2}}\right)\right)\right)}\]

    5. Simplified0.0

      \[\leadsto \log \color{blue}{\left(e^{\frac{-1}{2}} \cdot \left({x}^{4} \cdot \frac{351}{22400} + \left(1 + x \cdot \left(x \cdot \frac{9}{40}\right)\right)\right)\right)}\]

    6. Taylor expanded around 0 0.0

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

    7. Simplified0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.026253755058246165 \lor \neg \left(x \le 0.0282136986304838445\right):\\ \;\;\;\;\log \left(e^{\frac{x - \sin x}{x - \tan x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} + \left({x}^{4} \cdot \frac{-27}{2800} + x \cdot \left(x \cdot \frac{9}{40}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020182 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))