Average Error: 31.3 → 0.0
Time: 9.9s
Precision: 64
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.02600941637726964:\\ \;\;\;\;\log \left(e^{\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}}\right)\\ \mathbf{elif}\;x \le 0.026693962352464101:\\ \;\;\;\;\frac{9}{40} \cdot {x}^{2} - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \end{array}\]
\frac{x - \sin x}{x - \tan x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.02600941637726964:\\
\;\;\;\;\log \left(e^{\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}}\right)\\

\mathbf{elif}\;x \le 0.026693962352464101:\\
\;\;\;\;\frac{9}{40} \cdot {x}^{2} - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{x - \sin x}{x - \tan x}\\

\end{array}
double f(double x) {
        double r13813 = x;
        double r13814 = sin(r13813);
        double r13815 = r13813 - r13814;
        double r13816 = tan(r13813);
        double r13817 = r13813 - r13816;
        double r13818 = r13815 / r13817;
        return r13818;
}
↓
double f(double x) {
        double r13819 = x;
        double r13820 = -0.02600941637726964;
        bool r13821 = r13819 <= r13820;
        double r13822 = tan(r13819);
        double r13823 = r13819 - r13822;
        double r13824 = r13819 / r13823;
        double r13825 = sin(r13819);
        double r13826 = r13825 / r13823;
        double r13827 = r13824 - r13826;
        double r13828 = exp(r13827);
        double r13829 = log(r13828);
        double r13830 = 0.0266939623524641;
        bool r13831 = r13819 <= r13830;
        double r13832 = 0.225;
        double r13833 = 2.0;
        double r13834 = pow(r13819, r13833);
        double r13835 = r13832 * r13834;
        double r13836 = 0.009642857142857142;
        double r13837 = 4.0;
        double r13838 = pow(r13819, r13837);
        double r13839 = r13836 * r13838;
        double r13840 = 0.5;
        double r13841 = r13839 + r13840;
        double r13842 = r13835 - r13841;
        double r13843 = r13819 - r13825;
        double r13844 = r13843 / r13823;
        double r13845 = r13831 ? r13842 : r13844;
        double r13846 = r13821 ? r13829 : r13845;
        return r13846;
}

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 < -0.02600941637726964

    1. Initial program 0.1

      \[\frac{x - \sin x}{x - \tan x}\]
    2. Using strategy rm

    3. Applied div-sub0.1

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

    5. Applied add-log-exp0.2

      \[\leadsto \frac{x}{x - \tan x} - \color{blue}{\log \left(e^{\frac{\sin x}{x - \tan x}}\right)}\]

    6. Applied add-log-exp0.2

      \[\leadsto \color{blue}{\log \left(e^{\frac{x}{x - \tan x}}\right)} - \log \left(e^{\frac{\sin x}{x - \tan x}}\right)\]

    7. Applied diff-log0.2

      \[\leadsto \color{blue}{\log \left(\frac{e^{\frac{x}{x - \tan x}}}{e^{\frac{\sin x}{x - \tan x}}}\right)}\]

    8. Simplified0.1

      \[\leadsto \log \color{blue}{\left(e^{\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}}\right)}\]

    if -0.02600941637726964 < x < 0.0266939623524641

    1. Initial program 63.2

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

    2. Taylor expanded around 0 0.0

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

    if 0.0266939623524641 < x

    1. Initial program 0.1

      \[\frac{x - \sin x}{x - \tan x}\]
    2. Using strategy rm

    3. Applied div-sub0.1

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

    5. Applied sub-div0.1

      \[\leadsto \color{blue}{\frac{x - \sin x}{x - \tan x}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.02600941637726964:\\ \;\;\;\;\log \left(e^{\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}}\right)\\ \mathbf{elif}\;x \le 0.026693962352464101:\\ \;\;\;\;\frac{9}{40} \cdot {x}^{2} - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))