Average Error: 31.5 → 0.3
Time: 30.0s
Precision: 64
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2.446923987498581:\\ \;\;\;\;\left(\frac{\frac{\sin x}{\cos x}}{x} \cdot \frac{\frac{\sin x}{\cos x}}{x} + \left(\left(1 - \frac{\sin x}{x}\right) + \frac{\frac{\sin x}{x}}{\cos x}\right)\right) - \frac{\sin x}{x \cdot x} \cdot \frac{\sin x}{\cos x}\\ \mathbf{elif}\;x \le 2.4570066373596267:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\frac{9}{40} - \left(x \cdot x\right) \cdot \frac{27}{2800}\right) - \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{\sin x}{\cos x}}{x} \cdot \frac{\frac{\sin x}{\cos x}}{x} + \left(\left(1 - \frac{\sin x}{x}\right) + \frac{\frac{\sin x}{x}}{\cos x}\right)\right) - \frac{\sin x}{x \cdot x} \cdot \frac{\sin x}{\cos x}\\ \end{array}\]
\frac{x - \sin x}{x - \tan x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -2.446923987498581:\\
\;\;\;\;\left(\frac{\frac{\sin x}{\cos x}}{x} \cdot \frac{\frac{\sin x}{\cos x}}{x} + \left(\left(1 - \frac{\sin x}{x}\right) + \frac{\frac{\sin x}{x}}{\cos x}\right)\right) - \frac{\sin x}{x \cdot x} \cdot \frac{\sin x}{\cos x}\\

\mathbf{elif}\;x \le 2.4570066373596267:\\
\;\;\;\;\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}:\\
\;\;\;\;\left(\frac{\frac{\sin x}{\cos x}}{x} \cdot \frac{\frac{\sin x}{\cos x}}{x} + \left(\left(1 - \frac{\sin x}{x}\right) + \frac{\frac{\sin x}{x}}{\cos x}\right)\right) - \frac{\sin x}{x \cdot x} \cdot \frac{\sin x}{\cos x}\\

\end{array}
double f(double x) {
        double r593708 = x;
        double r593709 = sin(r593708);
        double r593710 = r593708 - r593709;
        double r593711 = tan(r593708);
        double r593712 = r593708 - r593711;
        double r593713 = r593710 / r593712;
        return r593713;
}
↓
double f(double x) {
        double r593714 = x;
        double r593715 = -2.446923987498581;
        bool r593716 = r593714 <= r593715;
        double r593717 = sin(r593714);
        double r593718 = cos(r593714);
        double r593719 = r593717 / r593718;
        double r593720 = r593719 / r593714;
        double r593721 = r593720 * r593720;
        double r593722 = 1.0;
        double r593723 = r593717 / r593714;
        double r593724 = r593722 - r593723;
        double r593725 = r593723 / r593718;
        double r593726 = r593724 + r593725;
        double r593727 = r593721 + r593726;
        double r593728 = r593714 * r593714;
        double r593729 = r593717 / r593728;
        double r593730 = r593729 * r593719;
        double r593731 = r593727 - r593730;
        double r593732 = 2.4570066373596267;
        bool r593733 = r593714 <= r593732;
        double r593734 = 0.225;
        double r593735 = 0.009642857142857142;
        double r593736 = r593728 * r593735;
        double r593737 = r593734 - r593736;
        double r593738 = r593728 * r593737;
        double r593739 = 0.5;
        double r593740 = r593738 - r593739;
        double r593741 = r593733 ? r593740 : r593731;
        double r593742 = r593716 ? r593731 : r593741;
        return r593742;
}

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 < -2.446923987498581 or 2.4570066373596267 < x

    1. Initial program 0.0

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

    2. Taylor expanded around inf 0.4

      \[\leadsto \color{blue}{\left(1 + \left(\frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2} \cdot {x}^{2}} + \frac{\sin x}{\cos x \cdot x}\right)\right) - \left(\frac{\sin x}{x} + \frac{{\left(\sin x\right)}^{2}}{\cos x \cdot {x}^{2}}\right)}\]

    3. Simplified0.4

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

    if -2.446923987498581 < x < 2.4570066373596267

    1. Initial program 62.4

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

    2. Taylor expanded around 0 0.2

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

    3. Simplified0.2

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.446923987498581:\\ \;\;\;\;\left(\frac{\frac{\sin x}{\cos x}}{x} \cdot \frac{\frac{\sin x}{\cos x}}{x} + \left(\left(1 - \frac{\sin x}{x}\right) + \frac{\frac{\sin x}{x}}{\cos x}\right)\right) - \frac{\sin x}{x \cdot x} \cdot \frac{\sin x}{\cos x}\\ \mathbf{elif}\;x \le 2.4570066373596267:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\frac{9}{40} - \left(x \cdot x\right) \cdot \frac{27}{2800}\right) - \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{\sin x}{\cos x}}{x} \cdot \frac{\frac{\sin x}{\cos x}}{x} + \left(\left(1 - \frac{\sin x}{x}\right) + \frac{\frac{\sin x}{x}}{\cos x}\right)\right) - \frac{\sin x}{x \cdot x} \cdot \frac{\sin x}{\cos x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019144 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))