Average Error: 31.6 → 0.2
Time: 27.0s
Precision: 64
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.03029376308050287305850822860975313233212:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \mathbf{elif}\;x \le 2.40127543424934719595853493956383317709:\\ \;\;\;\;\left(\log \left(e^{\frac{9}{40} \cdot {x}^{2}}\right) - \frac{27}{2800} \cdot {x}^{4}\right) - \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{\sin x}{x \cdot \cos x} + 1\right) + \frac{\sin x}{{x}^{2} \cdot \cos x} \cdot \left(\frac{\sin x}{\cos x} - \sin x\right)\right) - \frac{\sin x}{x}\\ \end{array}\]
\frac{x - \sin x}{x - \tan x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.03029376308050287305850822860975313233212:\\
\;\;\;\;\frac{x - \sin x}{x - \tan x}\\

\mathbf{elif}\;x \le 2.40127543424934719595853493956383317709:\\
\;\;\;\;\left(\log \left(e^{\frac{9}{40} \cdot {x}^{2}}\right) - \frac{27}{2800} \cdot {x}^{4}\right) - \frac{1}{2}\\

\mathbf{else}:\\
\;\;\;\;\left(\left(\frac{\sin x}{x \cdot \cos x} + 1\right) + \frac{\sin x}{{x}^{2} \cdot \cos x} \cdot \left(\frac{\sin x}{\cos x} - \sin x\right)\right) - \frac{\sin x}{x}\\

\end{array}
double f(double x) {
        double r19690 = x;
        double r19691 = sin(r19690);
        double r19692 = r19690 - r19691;
        double r19693 = tan(r19690);
        double r19694 = r19690 - r19693;
        double r19695 = r19692 / r19694;
        return r19695;
}
↓
double f(double x) {
        double r19696 = x;
        double r19697 = -0.030293763080502873;
        bool r19698 = r19696 <= r19697;
        double r19699 = sin(r19696);
        double r19700 = r19696 - r19699;
        double r19701 = tan(r19696);
        double r19702 = r19696 - r19701;
        double r19703 = r19700 / r19702;
        double r19704 = 2.401275434249347;
        bool r19705 = r19696 <= r19704;
        double r19706 = 0.225;
        double r19707 = 2.0;
        double r19708 = pow(r19696, r19707);
        double r19709 = r19706 * r19708;
        double r19710 = exp(r19709);
        double r19711 = log(r19710);
        double r19712 = 0.009642857142857142;
        double r19713 = 4.0;
        double r19714 = pow(r19696, r19713);
        double r19715 = r19712 * r19714;
        double r19716 = r19711 - r19715;
        double r19717 = 0.5;
        double r19718 = r19716 - r19717;
        double r19719 = cos(r19696);
        double r19720 = r19696 * r19719;
        double r19721 = r19699 / r19720;
        double r19722 = 1.0;
        double r19723 = r19721 + r19722;
        double r19724 = r19708 * r19719;
        double r19725 = r19699 / r19724;
        double r19726 = r19699 / r19719;
        double r19727 = r19726 - r19699;
        double r19728 = r19725 * r19727;
        double r19729 = r19723 + r19728;
        double r19730 = r19699 / r19696;
        double r19731 = r19729 - r19730;
        double r19732 = r19705 ? r19718 : r19731;
        double r19733 = r19698 ? r19703 : r19732;
        return r19733;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if x < -0.030293763080502873

    1. Initial program 0.0

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

    if -0.030293763080502873 < x < 2.401275434249347

    1. Initial program 63.1

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

    2. Taylor expanded around 0 0.1

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

    4. Applied associate--r+0.1

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

    6. Applied add-log-exp0.1

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

    if 2.401275434249347 < x

    1. Initial program 0.0

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

    2. Taylor expanded around inf 0.3

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

    3. Simplified0.3

      \[\leadsto \color{blue}{\left(\left(\frac{\sin x}{x \cdot \cos x} + 1\right) + \frac{\sin x}{{x}^{2} \cdot \cos x} \cdot \left(\frac{\sin x}{\cos x} - \sin x\right)\right) - \frac{\sin x}{x}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.03029376308050287305850822860975313233212:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \mathbf{elif}\;x \le 2.40127543424934719595853493956383317709:\\ \;\;\;\;\left(\log \left(e^{\frac{9}{40} \cdot {x}^{2}}\right) - \frac{27}{2800} \cdot {x}^{4}\right) - \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{\sin x}{x \cdot \cos x} + 1\right) + \frac{\sin x}{{x}^{2} \cdot \cos x} \cdot \left(\frac{\sin x}{\cos x} - \sin x\right)\right) - \frac{\sin x}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))