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

↓

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

double f(double x, double eps) {
        double r6234573 = x;
        double r6234574 = eps;
        double r6234575 = r6234573 + r6234574;
        double r6234576 = sin(r6234575);
        double r6234577 = sin(r6234573);
        double r6234578 = r6234576 - r6234577;
        return r6234578;
}

↓

double f(double x, double eps) {
        double r6234579 = eps;
        double r6234580 = -0.0020458699213328716;
        bool r6234581 = r6234579 <= r6234580;
        double r6234582 = x;
        double r6234583 = sin(r6234582);
        double r6234584 = cos(r6234579);
        double r6234585 = r6234583 * r6234584;
        double r6234586 = cos(r6234582);
        double r6234587 = sin(r6234579);
        double r6234588 = r6234586 * r6234587;
        double r6234589 = r6234585 + r6234588;
        double r6234590 = r6234589 - r6234583;
        double r6234591 = 2.414794756854271e-24;
        bool r6234592 = r6234579 <= r6234591;
        double r6234593 = 2.0;
        double r6234594 = r6234579 / r6234593;
        double r6234595 = sin(r6234594);
        double r6234596 = r6234582 + r6234579;
        double r6234597 = r6234596 + r6234582;
        double r6234598 = r6234597 / r6234593;
        double r6234599 = cos(r6234598);
        double r6234600 = r6234595 * r6234599;
        double r6234601 = r6234593 * r6234600;
        double r6234602 = r6234588 - r6234583;
        double r6234603 = r6234602 + r6234585;
        double r6234604 = r6234592 ? r6234601 : r6234603;
        double r6234605 = r6234581 ? r6234590 : r6234604;
        return r6234605;
}

\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