Average Error: 31.9 → 0.0
Time: 21.0s
Precision: 64
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.02911457298168511689806514652900659712031:\\ \;\;\;\;\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\\ \mathbf{elif}\;x \le 0.02897311180941740046956844878422998590395:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\frac{9}{40} - \left(x \cdot x\right) \cdot \frac{27}{2800}\right) - \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \end{array}\]
\frac{x - \sin x}{x - \tan x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.02911457298168511689806514652900659712031:\\
\;\;\;\;\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\\

\mathbf{elif}\;x \le 0.02897311180941740046956844878422998590395:\\
\;\;\;\;\left(x \cdot x\right) \cdot \left(\frac{9}{40} - \left(x \cdot x\right) \cdot \frac{27}{2800}\right) - \frac{1}{2}\\

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

\end{array}
double f(double x) {
        double r515877 = x;
        double r515878 = sin(r515877);
        double r515879 = r515877 - r515878;
        double r515880 = tan(r515877);
        double r515881 = r515877 - r515880;
        double r515882 = r515879 / r515881;
        return r515882;
}
↓
double f(double x) {
        double r515883 = x;
        double r515884 = -0.029114572981685117;
        bool r515885 = r515883 <= r515884;
        double r515886 = tan(r515883);
        double r515887 = r515883 - r515886;
        double r515888 = r515883 / r515887;
        double r515889 = sin(r515883);
        double r515890 = r515889 / r515887;
        double r515891 = r515888 - r515890;
        double r515892 = 0.0289731118094174;
        bool r515893 = r515883 <= r515892;
        double r515894 = r515883 * r515883;
        double r515895 = 0.225;
        double r515896 = 0.009642857142857142;
        double r515897 = r515894 * r515896;
        double r515898 = r515895 - r515897;
        double r515899 = r515894 * r515898;
        double r515900 = 0.5;
        double r515901 = r515899 - r515900;
        double r515902 = r515883 - r515889;
        double r515903 = r515902 / r515887;
        double r515904 = r515893 ? r515901 : r515903;
        double r515905 = r515885 ? r515891 : r515904;
        return r515905;
}

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 < -0.029114572981685117

    1. Initial program 0.1

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

    3. Applied div-sub0.1

      \[\leadsto \color{blue}{\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}}\]

    if -0.029114572981685117 < x < 0.0289731118094174

    1. Initial program 63.3

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

    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. Simplified0.0

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

    if 0.0289731118094174 < x

    1. Initial program 0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.02911457298168511689806514652900659712031:\\ \;\;\;\;\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\\ \mathbf{elif}\;x \le 0.02897311180941740046956844878422998590395:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(\frac{9}{40} - \left(x \cdot x\right) \cdot \frac{27}{2800}\right) - \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))