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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -4.95115446409742782 \cdot 10^{-9} \lor \neg \left(\varepsilon \le 2.17240038846879428 \cdot 10^{-10}\right):\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, -\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;\varepsilon \le -4.95115446409742782 \cdot 10^{-9} \lor \neg \left(\varepsilon \le 2.17240038846879428 \cdot 10^{-10}\right):\\
\;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, -\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\right)\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 r50697 = x;
        double r50698 = eps;
        double r50699 = r50697 + r50698;
        double r50700 = cos(r50699);
        double r50701 = cos(r50697);
        double r50702 = r50700 - r50701;
        return r50702;
}

↓

double f(double x, double eps) {
        double r50703 = eps;
        double r50704 = -4.951154464097428e-09;
        bool r50705 = r50703 <= r50704;
        double r50706 = 2.1724003884687943e-10;
        bool r50707 = r50703 <= r50706;
        double r50708 = !r50707;
        bool r50709 = r50705 || r50708;
        double r50710 = x;
        double r50711 = cos(r50710);
        double r50712 = cos(r50703);
        double r50713 = sin(r50710);
        double r50714 = sin(r50703);
        double r50715 = fma(r50713, r50714, r50711);
        double r50716 = expm1(r50715);
        double r50717 = log1p(r50716);
        double r50718 = -r50717;
        double r50719 = fma(r50711, r50712, r50718);
        double r50720 = 0.16666666666666666;
        double r50721 = 3.0;
        double r50722 = pow(r50710, r50721);
        double r50723 = r50720 * r50722;
        double r50724 = r50723 - r50710;
        double r50725 = 0.5;
        double r50726 = r50703 * r50725;
        double r50727 = r50724 - r50726;
        double r50728 = r50703 * r50727;
        double r50729 = r50709 ? r50719 : r50728;
        return r50729;
}

Derivation

Split input into 2 regimes
if eps < -4.951154464097428e-09 or 2.1724003884687943e-10 < eps
1. Initial program 30.8
  \[\cos \left(x + \varepsilon\right) - \cos x\]
2. Using strategy rm
3. Applied cos-sum1.4
  \[\leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
4. Applied associate--l-1.4
  \[\leadsto \color{blue}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)}\]
5. Simplified1.4
  \[\leadsto \cos x \cdot \cos \varepsilon - \color{blue}{\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)}\]
6. Using strategy rm
7. Applied fma-neg1.3
  \[\leadsto \color{blue}{\mathsf{fma}\left(\cos x, \cos \varepsilon, -\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\right)}\]
8. Using strategy rm
9. Applied log1p-expm1-u1.4
  \[\leadsto \mathsf{fma}\left(\cos x, \cos \varepsilon, -\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\right)\right)}\right)\]
if -4.951154464097428e-09 < eps < 2.1724003884687943e-10
1. Initial program 49.2
  \[\cos \left(x + \varepsilon\right) - \cos x\]
2. Taylor expanded around 0 31.8
  \[\leadsto \color{blue}{\frac{1}{6} \cdot \left({x}^{3} \cdot \varepsilon\right) - \left(x \cdot \varepsilon + \frac{1}{2} \cdot {\varepsilon}^{2}\right)}\]
3. Simplified31.8
  \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)}\]
Recombined 2 regimes into one program.
Final simplification15.8
\[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -4.95115446409742782 \cdot 10^{-9} \lor \neg \left(\varepsilon \le 2.17240038846879428 \cdot 10^{-10}\right):\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, -\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\ \end{array}\]

Error

Derivation

`if eps < -4.951154464097428e-09 or 2.1724003884687943e-10 < eps`

`if -4.951154464097428e-09 < eps < 2.1724003884687943e-10`

Reproduce