Average Error: 31.1 → 0.3
Time: 34.0s
Precision: 64
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2.4489045375766514:\\ \;\;\;\;\left(\frac{\frac{\sin x}{x}}{\cos x} \cdot \frac{\frac{\sin x}{x}}{\cos x} + \left(\left(1 + \frac{\frac{\sin x}{x}}{\cos x}\right) - \frac{\sin x}{x}\right)\right) - \frac{\frac{\sin x}{x}}{\cos x} \cdot \frac{\sin x}{x}\\ \mathbf{elif}\;x \le 2.4331115039478592:\\ \;\;\;\;\frac{-1}{2} + \left(x \cdot x\right) \cdot \left(\frac{9}{40} - \left(x \cdot x\right) \cdot \frac{27}{2800}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{\sin x}{x}}{\cos x} \cdot \frac{\frac{\sin x}{x}}{\cos x} + \left(\left(1 + \frac{\frac{\sin x}{x}}{\cos x}\right) - \frac{\sin x}{x}\right)\right) - \frac{\frac{\sin x}{x}}{\cos x} \cdot \frac{\sin x}{x}\\ \end{array}\]
\frac{x - \sin x}{x - \tan x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -2.4489045375766514:\\
\;\;\;\;\left(\frac{\frac{\sin x}{x}}{\cos x} \cdot \frac{\frac{\sin x}{x}}{\cos x} + \left(\left(1 + \frac{\frac{\sin x}{x}}{\cos x}\right) - \frac{\sin x}{x}\right)\right) - \frac{\frac{\sin x}{x}}{\cos x} \cdot \frac{\sin x}{x}\\

\mathbf{elif}\;x \le 2.4331115039478592:\\
\;\;\;\;\frac{-1}{2} + \left(x \cdot x\right) \cdot \left(\frac{9}{40} - \left(x \cdot x\right) \cdot \frac{27}{2800}\right)\\

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

\end{array}
double f(double x) {
        double r519572 = x;
        double r519573 = sin(r519572);
        double r519574 = r519572 - r519573;
        double r519575 = tan(r519572);
        double r519576 = r519572 - r519575;
        double r519577 = r519574 / r519576;
        return r519577;
}
↓
double f(double x) {
        double r519578 = x;
        double r519579 = -2.4489045375766514;
        bool r519580 = r519578 <= r519579;
        double r519581 = sin(r519578);
        double r519582 = r519581 / r519578;
        double r519583 = cos(r519578);
        double r519584 = r519582 / r519583;
        double r519585 = r519584 * r519584;
        double r519586 = 1.0;
        double r519587 = r519586 + r519584;
        double r519588 = r519587 - r519582;
        double r519589 = r519585 + r519588;
        double r519590 = r519584 * r519582;
        double r519591 = r519589 - r519590;
        double r519592 = 2.4331115039478592;
        bool r519593 = r519578 <= r519592;
        double r519594 = -0.5;
        double r519595 = r519578 * r519578;
        double r519596 = 0.225;
        double r519597 = 0.009642857142857142;
        double r519598 = r519595 * r519597;
        double r519599 = r519596 - r519598;
        double r519600 = r519595 * r519599;
        double r519601 = r519594 + r519600;
        double r519602 = r519593 ? r519601 : r519591;
        double r519603 = r519580 ? r519591 : r519602;
        return r519603;
}

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 < -2.4489045375766514 or 2.4331115039478592 < 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

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

    if -2.4489045375766514 < x < 2.4331115039478592

    1. Initial program 62.2

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

    2. Taylor expanded around 0 0.3

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

    3. Simplified0.3

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.4489045375766514:\\ \;\;\;\;\left(\frac{\frac{\sin x}{x}}{\cos x} \cdot \frac{\frac{\sin x}{x}}{\cos x} + \left(\left(1 + \frac{\frac{\sin x}{x}}{\cos x}\right) - \frac{\sin x}{x}\right)\right) - \frac{\frac{\sin x}{x}}{\cos x} \cdot \frac{\sin x}{x}\\ \mathbf{elif}\;x \le 2.4331115039478592:\\ \;\;\;\;\frac{-1}{2} + \left(x \cdot x\right) \cdot \left(\frac{9}{40} - \left(x \cdot x\right) \cdot \frac{27}{2800}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{\sin x}{x}}{\cos x} \cdot \frac{\frac{\sin x}{x}}{\cos x} + \left(\left(1 + \frac{\frac{\sin x}{x}}{\cos x}\right) - \frac{\sin x}{x}\right)\right) - \frac{\frac{\sin x}{x}}{\cos x} \cdot \frac{\sin x}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019141 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))