Average Error: 30.8 → 0.0
Time: 9.8s
Precision: 64
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.0289755674093477682 \lor \neg \left(x \le 0.028954196312268393\right):\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \mathbf{else}:\\ \;\;\;\;\frac{9}{40} \cdot {x}^{2} - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)\\ \end{array}\]
\frac{x - \sin x}{x - \tan x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.0289755674093477682 \lor \neg \left(x \le 0.028954196312268393\right):\\
\;\;\;\;\frac{x - \sin x}{x - \tan x}\\

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

\end{array}
double f(double x) {
        double r13448 = x;
        double r13449 = sin(r13448);
        double r13450 = r13448 - r13449;
        double r13451 = tan(r13448);
        double r13452 = r13448 - r13451;
        double r13453 = r13450 / r13452;
        return r13453;
}
↓
double f(double x) {
        double r13454 = x;
        double r13455 = -0.028975567409347768;
        bool r13456 = r13454 <= r13455;
        double r13457 = 0.028954196312268393;
        bool r13458 = r13454 <= r13457;
        double r13459 = !r13458;
        bool r13460 = r13456 || r13459;
        double r13461 = sin(r13454);
        double r13462 = r13454 - r13461;
        double r13463 = tan(r13454);
        double r13464 = r13454 - r13463;
        double r13465 = r13462 / r13464;
        double r13466 = 0.225;
        double r13467 = 2.0;
        double r13468 = pow(r13454, r13467);
        double r13469 = r13466 * r13468;
        double r13470 = 0.009642857142857142;
        double r13471 = 4.0;
        double r13472 = pow(r13454, r13471);
        double r13473 = r13470 * r13472;
        double r13474 = 0.5;
        double r13475 = r13473 + r13474;
        double r13476 = r13469 - r13475;
        double r13477 = r13460 ? r13465 : r13476;
        return r13477;
}

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 < -0.028975567409347768 or 0.028954196312268393 < x

    1. Initial program 0.0

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

    if -0.028975567409347768 < x < 0.028954196312268393

    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. Recombined 2 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0289755674093477682 \lor \neg \left(x \le 0.028954196312268393\right):\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \mathbf{else}:\\ \;\;\;\;\frac{9}{40} \cdot {x}^{2} - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))