\[\cos \left(x + \varepsilon\right) - \cos x\]

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -3.658449891821199522487853550423153592419 \cdot 10^{-6} \lor \neg \left(\varepsilon \le 3.130998694967744134270873183900629754817 \cdot 10^{-8}\right):\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, -\left(\sin x \cdot \sin \varepsilon + \cos x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\ \end{array}\]

\cos \left(x + \varepsilon\right) - \cos x

↓

\begin{array}{l}
\mathbf{if}\;\varepsilon \le -3.658449891821199522487853550423153592419 \cdot 10^{-6} \lor \neg \left(\varepsilon \le 3.130998694967744134270873183900629754817 \cdot 10^{-8}\right):\\
\;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, -\left(\sin x \cdot \sin \varepsilon + \cos x\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\

\end{array}

double f(double x, double eps) {
        double r58846 = x;
        double r58847 = eps;
        double r58848 = r58846 + r58847;
        double r58849 = cos(r58848);
        double r58850 = cos(r58846);
        double r58851 = r58849 - r58850;
        return r58851;
}

↓

double f(double x, double eps) {
        double r58852 = eps;
        double r58853 = -3.6584498918211995e-06;
        bool r58854 = r58852 <= r58853;
        double r58855 = 3.130998694967744e-08;
        bool r58856 = r58852 <= r58855;
        double r58857 = !r58856;
        bool r58858 = r58854 || r58857;
        double r58859 = x;
        double r58860 = cos(r58859);
        double r58861 = cos(r58852);
        double r58862 = sin(r58859);
        double r58863 = sin(r58852);
        double r58864 = r58862 * r58863;
        double r58865 = r58864 + r58860;
        double r58866 = -r58865;
        double r58867 = fma(r58860, r58861, r58866);
        double r58868 = 0.16666666666666666;
        double r58869 = 3.0;
        double r58870 = pow(r58859, r58869);
        double r58871 = r58868 * r58870;
        double r58872 = r58871 - r58859;
        double r58873 = 0.5;
        double r58874 = r58852 * r58873;
        double r58875 = r58872 - r58874;
        double r58876 = r58852 * r58875;
        double r58877 = r58858 ? r58867 : r58876;
        return r58877;
}

Derivation

Split input into 2 regimes
if eps < -3.6584498918211995e-06 or 3.130998694967744e-08 < eps
1. Initial program 30.1
  \[\cos \left(x + \varepsilon\right) - \cos x\]
2. Using strategy rm
3. Applied cos-sum1.1
  \[\leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
4. Applied associate--l-1.1
  \[\leadsto \color{blue}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)}\]
5. Simplified1.1
  \[\leadsto \cos x \cdot \cos \varepsilon - \color{blue}{\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)}\]
6. Using strategy rm
7. Applied fma-neg1.1
  \[\leadsto \color{blue}{\mathsf{fma}\left(\cos x, \cos \varepsilon, -\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\right)}\]
8. Using strategy rm
9. Applied fma-udef1.1
  \[\leadsto \mathsf{fma}\left(\cos x, \cos \varepsilon, -\color{blue}{\left(\sin x \cdot \sin \varepsilon + \cos x\right)}\right)\]
if -3.6584498918211995e-06 < eps < 3.130998694967744e-08
1. Initial program 49.7
  \[\cos \left(x + \varepsilon\right) - \cos x\]
2. Taylor expanded around 0 32.6
  \[\leadsto \color{blue}{\frac{1}{6} \cdot \left({x}^{3} \cdot \varepsilon\right) - \left(x \cdot \varepsilon + \frac{1}{2} \cdot {\varepsilon}^{2}\right)}\]
3. Simplified32.6
  \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)}\]
Recombined 2 regimes into one program.
Final simplification16.5
\[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -3.658449891821199522487853550423153592419 \cdot 10^{-6} \lor \neg \left(\varepsilon \le 3.130998694967744134270873183900629754817 \cdot 10^{-8}\right):\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, -\left(\sin x \cdot \sin \varepsilon + \cos x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\ \end{array}\]

Error

Derivation

`if eps < -3.6584498918211995e-06 or 3.130998694967744e-08 < eps`

`if -3.6584498918211995e-06 < eps < 3.130998694967744e-08`

Reproduce