Average Error: 31.2 → 0.0
Time: 29.8s
Precision: 64
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.03333148510890561:\\ \;\;\;\;\frac{1}{\frac{x - \frac{\sin x}{\cos x}}{x - \sin x}}\\ \mathbf{elif}\;x \le 0.029335377355220408:\\ \;\;\;\;\left(\frac{-1}{2} + x \cdot \left(x \cdot \frac{9}{40}\right)\right) + \frac{-27}{2800} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x - \frac{\sin x}{\cos x}}{x - \sin x}}\\ \end{array}\]
\frac{x - \sin x}{x - \tan x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.03333148510890561:\\
\;\;\;\;\frac{1}{\frac{x - \frac{\sin x}{\cos x}}{x - \sin x}}\\

\mathbf{elif}\;x \le 0.029335377355220408:\\
\;\;\;\;\left(\frac{-1}{2} + x \cdot \left(x \cdot \frac{9}{40}\right)\right) + \frac{-27}{2800} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{\frac{x - \frac{\sin x}{\cos x}}{x - \sin x}}\\

\end{array}
double f(double x) {
        double r903459 = x;
        double r903460 = sin(r903459);
        double r903461 = r903459 - r903460;
        double r903462 = tan(r903459);
        double r903463 = r903459 - r903462;
        double r903464 = r903461 / r903463;
        return r903464;
}
↓
double f(double x) {
        double r903465 = x;
        double r903466 = -0.03333148510890561;
        bool r903467 = r903465 <= r903466;
        double r903468 = 1.0;
        double r903469 = sin(r903465);
        double r903470 = cos(r903465);
        double r903471 = r903469 / r903470;
        double r903472 = r903465 - r903471;
        double r903473 = r903465 - r903469;
        double r903474 = r903472 / r903473;
        double r903475 = r903468 / r903474;
        double r903476 = 0.029335377355220408;
        bool r903477 = r903465 <= r903476;
        double r903478 = -0.5;
        double r903479 = 0.225;
        double r903480 = r903465 * r903479;
        double r903481 = r903465 * r903480;
        double r903482 = r903478 + r903481;
        double r903483 = -0.009642857142857142;
        double r903484 = r903465 * r903465;
        double r903485 = r903484 * r903484;
        double r903486 = r903483 * r903485;
        double r903487 = r903482 + r903486;
        double r903488 = r903477 ? r903487 : r903475;
        double r903489 = r903467 ? r903475 : r903488;
        return r903489;
}

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 < -0.03333148510890561 or 0.029335377355220408 < x

    1. Initial program 0.0

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

    2. Taylor expanded around inf 0.0

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

    4. Applied clear-num0.1

      \[\leadsto \color{blue}{\frac{1}{\frac{x - \frac{\sin x}{\cos x}}{x - \sin x}}}\]

    if -0.03333148510890561 < x < 0.029335377355220408

    1. Initial program 62.7

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

    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

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.03333148510890561:\\ \;\;\;\;\frac{1}{\frac{x - \frac{\sin x}{\cos x}}{x - \sin x}}\\ \mathbf{elif}\;x \le 0.029335377355220408:\\ \;\;\;\;\left(\frac{-1}{2} + x \cdot \left(x \cdot \frac{9}{40}\right)\right) + \frac{-27}{2800} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x - \frac{\sin x}{\cos x}}{x - \sin x}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))