Average Error: 31.7 → 0.0
Time: 9.3s
Precision: binary64
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.0294038700472425453 \lor \neg \left(x \le 0.027100292410791739\right):\\ \;\;\;\;\log \left(e^{\frac{x - \sin x}{x - \tan x}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{9}{40}\right) + \left({x}^{4} \cdot \frac{-27}{2800} + \frac{-1}{2}\right)\\ \end{array}\]
\frac{x - \sin x}{x - \tan x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.0294038700472425453 \lor \neg \left(x \le 0.027100292410791739\right):\\
\;\;\;\;\log \left(e^{\frac{x - \sin x}{x - \tan x}}\right)\\

\mathbf{else}:\\
\;\;\;\;x \cdot \left(x \cdot \frac{9}{40}\right) + \left({x}^{4} \cdot \frac{-27}{2800} + \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 <= -0.029403870047242545) || !(x <= 0.02710029241079174))) {
		VAR = ((double) log(((double) exp(((double) (((double) (x - ((double) sin(x)))) / ((double) (x - ((double) tan(x))))))))));
	} else {
		VAR = ((double) (((double) (x * ((double) (x * 0.225)))) + ((double) (((double) (((double) pow(x, 4.0)) * -0.009642857142857142)) + -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 < -0.0294038700472425453 or 0.027100292410791739 < 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 -0.0294038700472425453 < x < 0.027100292410791739

    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. Simplified0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0294038700472425453 \lor \neg \left(x \le 0.027100292410791739\right):\\ \;\;\;\;\log \left(e^{\frac{x - \sin x}{x - \tan x}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{9}{40}\right) + \left({x}^{4} \cdot \frac{-27}{2800} + \frac{-1}{2}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020179 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))