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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -9.8390408000405122 \cdot 10^{-17} \lor \neg \left(\varepsilon \le 1.8711149652135092 \cdot 10^{-6}\right):\\ \;\;\;\;\mathsf{fma}\left(\cos \varepsilon, \cos x, -\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left({\varepsilon}^{3} \cdot \frac{1}{24} - \mathsf{fma}\left(\frac{1}{2}, \varepsilon, x\right)\right)\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;\varepsilon \le -9.8390408000405122 \cdot 10^{-17} \lor \neg \left(\varepsilon \le 1.8711149652135092 \cdot 10^{-6}\right):\\
\;\;\;\;\mathsf{fma}\left(\cos \varepsilon, \cos x, -\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\right)\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\varepsilon \cdot \left({\varepsilon}^{3} \cdot \frac{1}{24} - \mathsf{fma}\left(\frac{1}{2}, \varepsilon, x\right)\right)\\

\end{array}

double f(double x, double eps) {
        double r61542 = x;
        double r61543 = eps;
        double r61544 = r61542 + r61543;
        double r61545 = cos(r61544);
        double r61546 = cos(r61542);
        double r61547 = r61545 - r61546;
        return r61547;
}

↓

double f(double x, double eps) {
        double r61548 = eps;
        double r61549 = -9.839040800040512e-17;
        bool r61550 = r61548 <= r61549;
        double r61551 = 1.8711149652135092e-06;
        bool r61552 = r61548 <= r61551;
        double r61553 = !r61552;
        bool r61554 = r61550 || r61553;
        double r61555 = cos(r61548);
        double r61556 = x;
        double r61557 = cos(r61556);
        double r61558 = sin(r61556);
        double r61559 = sin(r61548);
        double r61560 = fma(r61558, r61559, r61557);
        double r61561 = expm1(r61560);
        double r61562 = log1p(r61561);
        double r61563 = -r61562;
        double r61564 = fma(r61555, r61557, r61563);
        double r61565 = 3.0;
        double r61566 = pow(r61548, r61565);
        double r61567 = 0.041666666666666664;
        double r61568 = r61566 * r61567;
        double r61569 = 0.5;
        double r61570 = fma(r61569, r61548, r61556);
        double r61571 = r61568 - r61570;
        double r61572 = r61548 * r61571;
        double r61573 = r61554 ? r61564 : r61572;
        return r61573;
}

Derivation

Split input into 2 regimes
if eps < -9.839040800040512e-17 or 1.8711149652135092e-06 < eps
1. Initial program 30.2
  \[\cos \left(x + \varepsilon\right) - \cos x\]
2. Using strategy rm
3. Applied cos-sum1.6
  \[\leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
4. Taylor expanded around inf 1.6
  \[\leadsto \color{blue}{\cos \varepsilon \cdot \cos x - \left(\sin x \cdot \sin \varepsilon + \cos x\right)}\]
5. Simplified1.5
  \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \varepsilon, \cos x, -\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\right)}\]
6. Using strategy rm
7. Applied log1p-expm1-u1.6
  \[\leadsto \mathsf{fma}\left(\cos \varepsilon, \cos x, -\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\right)\right)}\right)\]
if -9.839040800040512e-17 < eps < 1.8711149652135092e-06
1. Initial program 49.1
  \[\cos \left(x + \varepsilon\right) - \cos x\]
2. Using strategy rm
3. Applied cos-sum48.9
  \[\leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
4. Taylor expanded around inf 48.9
  \[\leadsto \color{blue}{\cos \varepsilon \cdot \cos x - \left(\sin x \cdot \sin \varepsilon + \cos x\right)}\]
5. Simplified48.9
  \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \varepsilon, \cos x, -\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\right)}\]
6. Taylor expanded around 0 31.5
  \[\leadsto \color{blue}{\frac{1}{24} \cdot {\varepsilon}^{4} - \left(x \cdot \varepsilon + \frac{1}{2} \cdot {\varepsilon}^{2}\right)}\]
7. Simplified31.5
  \[\leadsto \color{blue}{\varepsilon \cdot \left({\varepsilon}^{3} \cdot \frac{1}{24} - \mathsf{fma}\left(\frac{1}{2}, \varepsilon, x\right)\right)}\]
Recombined 2 regimes into one program.
Final simplification16.3
\[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -9.8390408000405122 \cdot 10^{-17} \lor \neg \left(\varepsilon \le 1.8711149652135092 \cdot 10^{-6}\right):\\ \;\;\;\;\mathsf{fma}\left(\cos \varepsilon, \cos x, -\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left({\varepsilon}^{3} \cdot \frac{1}{24} - \mathsf{fma}\left(\frac{1}{2}, \varepsilon, x\right)\right)\\ \end{array}\]

Error

Derivation

`if eps < -9.839040800040512e-17 or 1.8711149652135092e-06 < eps`

`if -9.839040800040512e-17 < eps < 1.8711149652135092e-06`

Reproduce