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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -4.95115446409742782 \cdot 10^{-9}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \sqrt[3]{{\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^{3}}\\ \mathbf{elif}\;\varepsilon \le 2.17240038846879428 \cdot 10^{-10}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \log \left(e^{\sin x \cdot \sin \varepsilon + \cos x}\right)\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;\varepsilon \le -4.95115446409742782 \cdot 10^{-9}:\\
\;\;\;\;\cos x \cdot \cos \varepsilon - \sqrt[3]{{\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^{3}}\\

\mathbf{elif}\;\varepsilon \le 2.17240038846879428 \cdot 10^{-10}:\\
\;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\

\mathbf{else}:\\
\;\;\;\;\cos x \cdot \cos \varepsilon - \log \left(e^{\sin x \cdot \sin \varepsilon + \cos x}\right)\\

\end{array}

double f(double x, double eps) {
        double r60623 = x;
        double r60624 = eps;
        double r60625 = r60623 + r60624;
        double r60626 = cos(r60625);
        double r60627 = cos(r60623);
        double r60628 = r60626 - r60627;
        return r60628;
}

↓

double f(double x, double eps) {
        double r60629 = eps;
        double r60630 = -4.951154464097428e-09;
        bool r60631 = r60629 <= r60630;
        double r60632 = x;
        double r60633 = cos(r60632);
        double r60634 = cos(r60629);
        double r60635 = r60633 * r60634;
        double r60636 = sin(r60632);
        double r60637 = sin(r60629);
        double r60638 = r60636 * r60637;
        double r60639 = r60638 + r60633;
        double r60640 = 3.0;
        double r60641 = pow(r60639, r60640);
        double r60642 = cbrt(r60641);
        double r60643 = r60635 - r60642;
        double r60644 = 2.1724003884687943e-10;
        bool r60645 = r60629 <= r60644;
        double r60646 = 0.16666666666666666;
        double r60647 = pow(r60632, r60640);
        double r60648 = r60646 * r60647;
        double r60649 = r60648 - r60632;
        double r60650 = 0.5;
        double r60651 = r60629 * r60650;
        double r60652 = r60649 - r60651;
        double r60653 = r60629 * r60652;
        double r60654 = exp(r60639);
        double r60655 = log(r60654);
        double r60656 = r60635 - r60655;
        double r60657 = r60645 ? r60653 : r60656;
        double r60658 = r60631 ? r60643 : r60657;
        return r60658;
}

Derivation

Split input into 3 regimes
if eps < -4.951154464097428e-09
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. Using strategy rm
6. Applied add-cbrt-cube1.5
  \[\leadsto \cos x \cdot \cos \varepsilon - \color{blue}{\sqrt[3]{\left(\left(\sin x \cdot \sin \varepsilon + \cos x\right) \cdot \left(\sin x \cdot \sin \varepsilon + \cos x\right)\right) \cdot \left(\sin x \cdot \sin \varepsilon + \cos x\right)}}\]
7. Simplified1.5
  \[\leadsto \cos x \cdot \cos \varepsilon - \sqrt[3]{\color{blue}{{\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^{3}}}\]
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
  \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)}\]
if 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. Using strategy rm
6. Applied add-log-exp1.5
  \[\leadsto \cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \color{blue}{\log \left(e^{\cos x}\right)}\right)\]
7. Applied add-log-exp1.5
  \[\leadsto \cos x \cdot \cos \varepsilon - \left(\color{blue}{\log \left(e^{\sin x \cdot \sin \varepsilon}\right)} + \log \left(e^{\cos x}\right)\right)\]
8. Applied sum-log1.6
  \[\leadsto \cos x \cdot \cos \varepsilon - \color{blue}{\log \left(e^{\sin x \cdot \sin \varepsilon} \cdot e^{\cos x}\right)}\]
9. Simplified1.5
  \[\leadsto \cos x \cdot \cos \varepsilon - \log \color{blue}{\left(e^{\sin x \cdot \sin \varepsilon + \cos x}\right)}\]
Recombined 3 regimes into one program.
Final simplification15.9
\[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -4.95115446409742782 \cdot 10^{-9}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \sqrt[3]{{\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^{3}}\\ \mathbf{elif}\;\varepsilon \le 2.17240038846879428 \cdot 10^{-10}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \log \left(e^{\sin x \cdot \sin \varepsilon + \cos x}\right)\\ \end{array}\]

Error

Try it out

Derivation

`if eps < -4.951154464097428e-09`

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

`if 2.1724003884687943e-10 < eps`

Reproduce

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