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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -0.0020458699213328716:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 2.414794756854271 \cdot 10^{-24}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\ \end{array}\]

double f(double x, double eps) {
        double r10340127 = x;
        double r10340128 = eps;
        double r10340129 = r10340127 + r10340128;
        double r10340130 = sin(r10340129);
        double r10340131 = sin(r10340127);
        double r10340132 = r10340130 - r10340131;
        return r10340132;
}

↓

double f(double x, double eps) {
        double r10340133 = eps;
        double r10340134 = -0.0020458699213328716;
        bool r10340135 = r10340133 <= r10340134;
        double r10340136 = x;
        double r10340137 = sin(r10340136);
        double r10340138 = cos(r10340133);
        double r10340139 = r10340137 * r10340138;
        double r10340140 = cos(r10340136);
        double r10340141 = sin(r10340133);
        double r10340142 = r10340140 * r10340141;
        double r10340143 = r10340139 + r10340142;
        double r10340144 = r10340143 - r10340137;
        double r10340145 = 2.414794756854271e-24;
        bool r10340146 = r10340133 <= r10340145;
        double r10340147 = 2.0;
        double r10340148 = r10340133 / r10340147;
        double r10340149 = sin(r10340148);
        double r10340150 = r10340136 + r10340133;
        double r10340151 = r10340150 + r10340136;
        double r10340152 = r10340151 / r10340147;
        double r10340153 = cos(r10340152);
        double r10340154 = r10340149 * r10340153;
        double r10340155 = r10340147 * r10340154;
        double r10340156 = r10340142 - r10340137;
        double r10340157 = r10340156 + r10340139;
        double r10340158 = r10340146 ? r10340155 : r10340157;
        double r10340159 = r10340135 ? r10340144 : r10340158;
        return r10340159;
}

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

↓

\begin{array}{l}
\mathbf{if}\;\varepsilon \le -0.0020458699213328716:\\
\;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\

\mathbf{elif}\;\varepsilon \le 2.414794756854271 \cdot 10^{-24}:\\
\;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\

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

\end{array}

Original	36.7
Target	15.0
Herbie	0.7

Derivation

Split input into 3 regimes
if eps < -0.0020458699213328716
1. Initial program 30.4
  \[\sin \left(x + \varepsilon\right) - \sin x\]
2. Using strategy rm
3. Applied sin-sum0.4
  \[\leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x\]
if -0.0020458699213328716 < eps < 2.414794756854271e-24
1. Initial program 44.1
  \[\sin \left(x + \varepsilon\right) - \sin x\]
2. Using strategy rm
3. Applied diff-sin44.1
  \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)}\]
4. Simplified0.4
  \[\leadsto 2 \cdot \color{blue}{\left(\cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)}\]
if 2.414794756854271e-24 < eps
1. Initial program 29.3
  \[\sin \left(x + \varepsilon\right) - \sin x\]
2. Using strategy rm
3. Applied sin-sum1.5
  \[\leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x\]
4. Applied associate--l+1.5
  \[\leadsto \color{blue}{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)}\]
Recombined 3 regimes into one program.
Final simplification0.7
\[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -0.0020458699213328716:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 2.414794756854271 \cdot 10^{-24}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\ \end{array}\]

Error

Target

Derivation

`if eps < -0.0020458699213328716`

`if -0.0020458699213328716 < eps < 2.414794756854271e-24`

`if 2.414794756854271e-24 < eps`

Reproduce