Average Error: 31.4 → 0.0
Time: 24.3s
Precision: 64
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.02965602329990057323128027633174497168511:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \mathbf{elif}\;x \le 0.02964489669427243165311658401606109691784:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\frac{9}{40} - \left(x \cdot x\right) \cdot \frac{27}{2800}\right) - \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \end{array}\]
\frac{x - \sin x}{x - \tan x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.02965602329990057323128027633174497168511:\\
\;\;\;\;\frac{x - \sin x}{x - \tan x}\\

\mathbf{elif}\;x \le 0.02964489669427243165311658401606109691784:\\
\;\;\;\;\left(x \cdot x\right) \cdot \left(\frac{9}{40} - \left(x \cdot x\right) \cdot \frac{27}{2800}\right) - \frac{1}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{x - \sin x}{x - \tan x}\\

\end{array}
double f(double x) {
        double r1095879 = x;
        double r1095880 = sin(r1095879);
        double r1095881 = r1095879 - r1095880;
        double r1095882 = tan(r1095879);
        double r1095883 = r1095879 - r1095882;
        double r1095884 = r1095881 / r1095883;
        return r1095884;
}
↓
double f(double x) {
        double r1095885 = x;
        double r1095886 = -0.029656023299900573;
        bool r1095887 = r1095885 <= r1095886;
        double r1095888 = sin(r1095885);
        double r1095889 = r1095885 - r1095888;
        double r1095890 = tan(r1095885);
        double r1095891 = r1095885 - r1095890;
        double r1095892 = r1095889 / r1095891;
        double r1095893 = 0.02964489669427243;
        bool r1095894 = r1095885 <= r1095893;
        double r1095895 = r1095885 * r1095885;
        double r1095896 = 0.225;
        double r1095897 = 0.009642857142857142;
        double r1095898 = r1095895 * r1095897;
        double r1095899 = r1095896 - r1095898;
        double r1095900 = r1095895 * r1095899;
        double r1095901 = 0.5;
        double r1095902 = r1095900 - r1095901;
        double r1095903 = r1095894 ? r1095902 : r1095892;
        double r1095904 = r1095887 ? r1095892 : r1095903;
        return r1095904;
}

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 < -0.029656023299900573 or 0.02964489669427243 < x

    1. Initial program 0.0

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

    3. Applied div-sub0.0

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

    5. Applied sub-div0.0

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

    if -0.029656023299900573 < x < 0.02964489669427243

    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. Simplified0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.02965602329990057323128027633174497168511:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \mathbf{elif}\;x \le 0.02964489669427243165311658401606109691784:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\frac{9}{40} - \left(x \cdot x\right) \cdot \frac{27}{2800}\right) - \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))