Average Error: 31.7 → 0.2
Time: 15.6s
Precision: 64
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.0269592498251367309:\\ \;\;\;\;\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\\ \mathbf{elif}\;x \le 2.4349587284165919:\\ \;\;\;\;\frac{9}{40} \cdot {x}^{2} - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sin x}{x \cdot \cos x} + \left(\frac{{\left(\sin x\right)}^{2}}{{x}^{2} \cdot {\left(\cos x\right)}^{2}} + 1\right)\right) - \left(\frac{\sin x}{x} + \frac{{\left(\sin x\right)}^{2}}{{x}^{2} \cdot \cos x}\right)\\ \end{array}\]
\frac{x - \sin x}{x - \tan x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.0269592498251367309:\\
\;\;\;\;\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\\

\mathbf{elif}\;x \le 2.4349587284165919:\\
\;\;\;\;\frac{9}{40} \cdot {x}^{2} - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)\\

\mathbf{else}:\\
\;\;\;\;\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)\\

\end{array}
double f(double x) {
        double r13113 = x;
        double r13114 = sin(r13113);
        double r13115 = r13113 - r13114;
        double r13116 = tan(r13113);
        double r13117 = r13113 - r13116;
        double r13118 = r13115 / r13117;
        return r13118;
}
↓
double f(double x) {
        double r13119 = x;
        double r13120 = -0.02695924982513673;
        bool r13121 = r13119 <= r13120;
        double r13122 = tan(r13119);
        double r13123 = r13119 - r13122;
        double r13124 = r13119 / r13123;
        double r13125 = sin(r13119);
        double r13126 = r13125 / r13123;
        double r13127 = r13124 - r13126;
        double r13128 = 2.434958728416592;
        bool r13129 = r13119 <= r13128;
        double r13130 = 0.225;
        double r13131 = 2.0;
        double r13132 = pow(r13119, r13131);
        double r13133 = r13130 * r13132;
        double r13134 = 0.009642857142857142;
        double r13135 = 4.0;
        double r13136 = pow(r13119, r13135);
        double r13137 = r13134 * r13136;
        double r13138 = 0.5;
        double r13139 = r13137 + r13138;
        double r13140 = r13133 - r13139;
        double r13141 = cos(r13119);
        double r13142 = r13119 * r13141;
        double r13143 = r13125 / r13142;
        double r13144 = pow(r13125, r13131);
        double r13145 = pow(r13141, r13131);
        double r13146 = r13132 * r13145;
        double r13147 = r13144 / r13146;
        double r13148 = 1.0;
        double r13149 = r13147 + r13148;
        double r13150 = r13143 + r13149;
        double r13151 = r13125 / r13119;
        double r13152 = r13132 * r13141;
        double r13153 = r13144 / r13152;
        double r13154 = r13151 + r13153;
        double r13155 = r13150 - r13154;
        double r13156 = r13129 ? r13140 : r13155;
        double r13157 = r13121 ? r13127 : r13156;
        return r13157;
}

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 < -0.02695924982513673

    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 -0.02695924982513673 < x < 2.434958728416592

    1. Initial program 63.0

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

    2. Taylor expanded around 0 0.1

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

    if 2.434958728416592 < 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. Recombined 3 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0269592498251367309:\\ \;\;\;\;\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\\ \mathbf{elif}\;x \le 2.4349587284165919:\\ \;\;\;\;\frac{9}{40} \cdot {x}^{2} - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sin x}{x \cdot \cos x} + \left(\frac{{\left(\sin x\right)}^{2}}{{x}^{2} \cdot {\left(\cos x\right)}^{2}} + 1\right)\right) - \left(\frac{\sin x}{x} + \frac{{\left(\sin x\right)}^{2}}{{x}^{2} \cdot \cos x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020044 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))