Average Error: 31.1 → 0.3
Time: 2.3m
Precision: 64
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2.447814972836657609889243758516386151314:\\ \;\;\;\;\left(\frac{\frac{\sin x}{x}}{\cos x} \cdot \frac{\frac{\sin x}{x}}{\cos x} - \left(\frac{\sin x}{x} + \frac{\frac{\sin x}{x} \cdot \frac{\sin x}{x}}{\cos x}\right)\right) + \left(1 + \frac{\frac{\sin x}{x}}{\cos x}\right)\\ \mathbf{elif}\;x \le 2.45325061863942162077023567690048366785:\\ \;\;\;\;x \cdot \left(x \cdot \frac{9}{40}\right) + \left(\frac{-27}{2800} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) - \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{\sin x}{x}}{\cos x} \cdot \frac{\frac{\sin x}{x}}{\cos x} - \left(\frac{\sin x}{x} + \frac{\frac{\sin x}{x} \cdot \frac{\sin x}{x}}{\cos x}\right)\right) + \left(1 + \frac{\frac{\sin x}{x}}{\cos x}\right)\\ \end{array}\]
\frac{x - \sin x}{x - \tan x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -2.447814972836657609889243758516386151314:\\
\;\;\;\;\left(\frac{\frac{\sin x}{x}}{\cos x} \cdot \frac{\frac{\sin x}{x}}{\cos x} - \left(\frac{\sin x}{x} + \frac{\frac{\sin x}{x} \cdot \frac{\sin x}{x}}{\cos x}\right)\right) + \left(1 + \frac{\frac{\sin x}{x}}{\cos x}\right)\\

\mathbf{elif}\;x \le 2.45325061863942162077023567690048366785:\\
\;\;\;\;x \cdot \left(x \cdot \frac{9}{40}\right) + \left(\frac{-27}{2800} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) - \frac{1}{2}\right)\\

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

\end{array}
double f(double x) {
        double r827153 = x;
        double r827154 = sin(r827153);
        double r827155 = r827153 - r827154;
        double r827156 = tan(r827153);
        double r827157 = r827153 - r827156;
        double r827158 = r827155 / r827157;
        return r827158;
}
↓
double f(double x) {
        double r827159 = x;
        double r827160 = -2.4478149728366576;
        bool r827161 = r827159 <= r827160;
        double r827162 = sin(r827159);
        double r827163 = r827162 / r827159;
        double r827164 = cos(r827159);
        double r827165 = r827163 / r827164;
        double r827166 = r827165 * r827165;
        double r827167 = r827163 * r827163;
        double r827168 = r827167 / r827164;
        double r827169 = r827163 + r827168;
        double r827170 = r827166 - r827169;
        double r827171 = 1.0;
        double r827172 = r827171 + r827165;
        double r827173 = r827170 + r827172;
        double r827174 = 2.4532506186394216;
        bool r827175 = r827159 <= r827174;
        double r827176 = 0.225;
        double r827177 = r827159 * r827176;
        double r827178 = r827159 * r827177;
        double r827179 = -0.009642857142857142;
        double r827180 = r827159 * r827159;
        double r827181 = r827180 * r827180;
        double r827182 = r827179 * r827181;
        double r827183 = 0.5;
        double r827184 = r827182 - r827183;
        double r827185 = r827178 + r827184;
        double r827186 = r827175 ? r827185 : r827173;
        double r827187 = r827161 ? r827173 : r827186;
        return r827187;
}

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 < -2.4478149728366576 or 2.4532506186394216 < x

    1. Initial program 0.0

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

    2. Taylor expanded around inf 0.4

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

    3. Simplified0.4

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

    if -2.4478149728366576 < x < 2.4532506186394216

    1. Initial program 62.8

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

    2. Taylor expanded around 0 0.2

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

    3. Simplified0.2

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.447814972836657609889243758516386151314:\\ \;\;\;\;\left(\frac{\frac{\sin x}{x}}{\cos x} \cdot \frac{\frac{\sin x}{x}}{\cos x} - \left(\frac{\sin x}{x} + \frac{\frac{\sin x}{x} \cdot \frac{\sin x}{x}}{\cos x}\right)\right) + \left(1 + \frac{\frac{\sin x}{x}}{\cos x}\right)\\ \mathbf{elif}\;x \le 2.45325061863942162077023567690048366785:\\ \;\;\;\;x \cdot \left(x \cdot \frac{9}{40}\right) + \left(\frac{-27}{2800} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) - \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{\sin x}{x}}{\cos x} \cdot \frac{\frac{\sin x}{x}}{\cos x} - \left(\frac{\sin x}{x} + \frac{\frac{\sin x}{x} \cdot \frac{\sin x}{x}}{\cos x}\right)\right) + \left(1 + \frac{\frac{\sin x}{x}}{\cos x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))