Average Error: 31.3 → 0.3
Time: 37.9s
Precision: 64
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2.475250416396811647956610613618977367878:\\ \;\;\;\;\left(\frac{\frac{\sin x}{x}}{\cos x} \cdot \frac{\frac{\sin x}{x}}{\cos x} + \left(\frac{\frac{\sin x}{x}}{\cos x} - \frac{\sin x}{x}\right)\right) - \left(\frac{\frac{\sin x \cdot \sin x}{\cos x}}{x \cdot x} - 1\right)\\ \mathbf{elif}\;x \le 2.408727669093524426102703728247433900833:\\ \;\;\;\;x \cdot \left(x \cdot \frac{9}{40}\right) - \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{27}{2800}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{\sin x}{x}}{\cos x} \cdot \frac{\frac{\sin x}{x}}{\cos x} + \left(\frac{\frac{\sin x}{x}}{\cos x} - \frac{\sin x}{x}\right)\right) - \left(\frac{\frac{\sin x \cdot \sin x}{\cos x}}{x \cdot x} - 1\right)\\ \end{array}\]
\frac{x - \sin x}{x - \tan x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -2.475250416396811647956610613618977367878:\\
\;\;\;\;\left(\frac{\frac{\sin x}{x}}{\cos x} \cdot \frac{\frac{\sin x}{x}}{\cos x} + \left(\frac{\frac{\sin x}{x}}{\cos x} - \frac{\sin x}{x}\right)\right) - \left(\frac{\frac{\sin x \cdot \sin x}{\cos x}}{x \cdot x} - 1\right)\\

\mathbf{elif}\;x \le 2.408727669093524426102703728247433900833:\\
\;\;\;\;x \cdot \left(x \cdot \frac{9}{40}\right) - \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{27}{2800}\right)\right)\\

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

\end{array}
double f(double x) {
        double r602749 = x;
        double r602750 = sin(r602749);
        double r602751 = r602749 - r602750;
        double r602752 = tan(r602749);
        double r602753 = r602749 - r602752;
        double r602754 = r602751 / r602753;
        return r602754;
}
↓
double f(double x) {
        double r602755 = x;
        double r602756 = -2.4752504163968116;
        bool r602757 = r602755 <= r602756;
        double r602758 = sin(r602755);
        double r602759 = r602758 / r602755;
        double r602760 = cos(r602755);
        double r602761 = r602759 / r602760;
        double r602762 = r602761 * r602761;
        double r602763 = r602761 - r602759;
        double r602764 = r602762 + r602763;
        double r602765 = r602758 * r602758;
        double r602766 = r602765 / r602760;
        double r602767 = r602755 * r602755;
        double r602768 = r602766 / r602767;
        double r602769 = 1.0;
        double r602770 = r602768 - r602769;
        double r602771 = r602764 - r602770;
        double r602772 = 2.4087276690935244;
        bool r602773 = r602755 <= r602772;
        double r602774 = 0.225;
        double r602775 = r602755 * r602774;
        double r602776 = r602755 * r602775;
        double r602777 = 0.5;
        double r602778 = 0.009642857142857142;
        double r602779 = r602767 * r602778;
        double r602780 = r602767 * r602779;
        double r602781 = r602777 + r602780;
        double r602782 = r602776 - r602781;
        double r602783 = r602773 ? r602782 : r602771;
        double r602784 = r602757 ? r602771 : r602783;
        return r602784;
}

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 < -2.4752504163968116 or 2.4087276690935244 < 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

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

    if -2.4752504163968116 < x < 2.4087276690935244

    1. Initial program 62.8

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

    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

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.475250416396811647956610613618977367878:\\ \;\;\;\;\left(\frac{\frac{\sin x}{x}}{\cos x} \cdot \frac{\frac{\sin x}{x}}{\cos x} + \left(\frac{\frac{\sin x}{x}}{\cos x} - \frac{\sin x}{x}\right)\right) - \left(\frac{\frac{\sin x \cdot \sin x}{\cos x}}{x \cdot x} - 1\right)\\ \mathbf{elif}\;x \le 2.408727669093524426102703728247433900833:\\ \;\;\;\;x \cdot \left(x \cdot \frac{9}{40}\right) - \left(\frac{1}{2} + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{27}{2800}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{\sin x}{x}}{\cos x} \cdot \frac{\frac{\sin x}{x}}{\cos x} + \left(\frac{\frac{\sin x}{x}}{\cos x} - \frac{\sin x}{x}\right)\right) - \left(\frac{\frac{\sin x \cdot \sin x}{\cos x}}{x \cdot x} - 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))