Average Error: 31.4 → 0.3
Time: 23.9s
Precision: 64
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2.422154288730760107739570230478420853615 \lor \neg \left(x \le 2.458140845788107320402104960521683096886\right):\\ \;\;\;\;\left(\frac{\sin x}{x \cdot \cos x} + \left(\frac{{\left(\sin x\right)}^{2}}{{x}^{2} \cdot {\left(\cos x\right)}^{2}} + 1\right)\right) - \left(\frac{\sin x}{x} + \frac{{\left(\sin x\right)}^{2}}{{x}^{2} \cdot \cos x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{9}{40} \cdot {x}^{2} - \frac{27}{2800} \cdot {x}^{4}\right) - \frac{1}{2}\\ \end{array}\]
\frac{x - \sin x}{x - \tan x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -2.422154288730760107739570230478420853615 \lor \neg \left(x \le 2.458140845788107320402104960521683096886\right):\\
\;\;\;\;\left(\frac{\sin x}{x \cdot \cos x} + \left(\frac{{\left(\sin x\right)}^{2}}{{x}^{2} \cdot {\left(\cos x\right)}^{2}} + 1\right)\right) - \left(\frac{\sin x}{x} + \frac{{\left(\sin x\right)}^{2}}{{x}^{2} \cdot \cos x}\right)\\

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

\end{array}
double f(double x) {
        double r19539 = x;
        double r19540 = sin(r19539);
        double r19541 = r19539 - r19540;
        double r19542 = tan(r19539);
        double r19543 = r19539 - r19542;
        double r19544 = r19541 / r19543;
        return r19544;
}
↓
double f(double x) {
        double r19545 = x;
        double r19546 = -2.42215428873076;
        bool r19547 = r19545 <= r19546;
        double r19548 = 2.4581408457881073;
        bool r19549 = r19545 <= r19548;
        double r19550 = !r19549;
        bool r19551 = r19547 || r19550;
        double r19552 = sin(r19545);
        double r19553 = cos(r19545);
        double r19554 = r19545 * r19553;
        double r19555 = r19552 / r19554;
        double r19556 = 2.0;
        double r19557 = pow(r19552, r19556);
        double r19558 = pow(r19545, r19556);
        double r19559 = pow(r19553, r19556);
        double r19560 = r19558 * r19559;
        double r19561 = r19557 / r19560;
        double r19562 = 1.0;
        double r19563 = r19561 + r19562;
        double r19564 = r19555 + r19563;
        double r19565 = r19552 / r19545;
        double r19566 = r19558 * r19553;
        double r19567 = r19557 / r19566;
        double r19568 = r19565 + r19567;
        double r19569 = r19564 - r19568;
        double r19570 = 0.225;
        double r19571 = r19570 * r19558;
        double r19572 = 0.009642857142857142;
        double r19573 = 4.0;
        double r19574 = pow(r19545, r19573);
        double r19575 = r19572 * r19574;
        double r19576 = r19571 - r19575;
        double r19577 = 0.5;
        double r19578 = r19576 - r19577;
        double r19579 = r19551 ? r19569 : r19578;
        return r19579;
}

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 < -2.42215428873076 or 2.4581408457881073 < x

    1. Initial program 0.0

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

    2. Taylor expanded around inf 0.4

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

    if -2.42215428873076 < x < 2.4581408457881073

    1. Initial program 62.7

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

    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. Using strategy rm

    4. Applied associate--r+0.2

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.422154288730760107739570230478420853615 \lor \neg \left(x \le 2.458140845788107320402104960521683096886\right):\\ \;\;\;\;\left(\frac{\sin x}{x \cdot \cos x} + \left(\frac{{\left(\sin x\right)}^{2}}{{x}^{2} \cdot {\left(\cos x\right)}^{2}} + 1\right)\right) - \left(\frac{\sin x}{x} + \frac{{\left(\sin x\right)}^{2}}{{x}^{2} \cdot \cos x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{9}{40} \cdot {x}^{2} - \frac{27}{2800} \cdot {x}^{4}\right) - \frac{1}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019297 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))