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

↓

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

\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 r50148 = x;
        double r50149 = eps;
        double r50150 = r50148 + r50149;
        double r50151 = cos(r50150);
        double r50152 = cos(r50148);
        double r50153 = r50151 - r50152;
        return r50153;
}

↓

double f(double x, double eps) {
        double r50154 = eps;
        double r50155 = -4.951154464097428e-09;
        bool r50156 = r50154 <= r50155;
        double r50157 = x;
        double r50158 = cos(r50157);
        double r50159 = cos(r50154);
        double r50160 = r50158 * r50159;
        double r50161 = sin(r50157);
        double r50162 = sin(r50154);
        double r50163 = r50161 * r50162;
        double r50164 = r50163 + r50158;
        double r50165 = 3.0;
        double r50166 = pow(r50164, r50165);
        double r50167 = cbrt(r50166);
        double r50168 = r50160 - r50167;
        double r50169 = 2.1724003884687943e-10;
        bool r50170 = r50154 <= r50169;
        double r50171 = 0.16666666666666666;
        double r50172 = pow(r50157, r50165);
        double r50173 = r50171 * r50172;
        double r50174 = r50173 - r50157;
        double r50175 = 0.5;
        double r50176 = r50154 * r50175;
        double r50177 = r50174 - r50176;
        double r50178 = r50154 * r50177;
        double r50179 = exp(r50164);
        double r50180 = log(r50179);
        double r50181 = r50160 - r50180;
        double r50182 = r50170 ? r50178 : r50181;
        double r50183 = r50156 ? r50168 : r50182;
        return r50183;
}

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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