Average Error: 37.2 → 0.6
Time: 18.5s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -65.72641980047264:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 9.625760540766168 \cdot 10^{-09}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\ \end{array}\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -65.72641980047264:\\
\;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\

\mathbf{elif}\;\varepsilon \le 9.625760540766168 \cdot 10^{-09}:\\
\;\;\;\;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 r7492722 = x;
        double r7492723 = eps;
        double r7492724 = r7492722 + r7492723;
        double r7492725 = sin(r7492724);
        double r7492726 = sin(r7492722);
        double r7492727 = r7492725 - r7492726;
        return r7492727;
}
↓
double f(double x, double eps) {
        double r7492728 = eps;
        double r7492729 = -65.72641980047264;
        bool r7492730 = r7492728 <= r7492729;
        double r7492731 = x;
        double r7492732 = sin(r7492731);
        double r7492733 = cos(r7492728);
        double r7492734 = r7492732 * r7492733;
        double r7492735 = cos(r7492731);
        double r7492736 = sin(r7492728);
        double r7492737 = r7492735 * r7492736;
        double r7492738 = r7492734 + r7492737;
        double r7492739 = r7492738 - r7492732;
        double r7492740 = 9.625760540766168e-09;
        bool r7492741 = r7492728 <= r7492740;
        double r7492742 = 2.0;
        double r7492743 = r7492728 / r7492742;
        double r7492744 = sin(r7492743);
        double r7492745 = r7492731 + r7492728;
        double r7492746 = r7492745 + r7492731;
        double r7492747 = r7492746 / r7492742;
        double r7492748 = cos(r7492747);
        double r7492749 = r7492744 * r7492748;
        double r7492750 = r7492742 * r7492749;
        double r7492751 = r7492737 - r7492732;
        double r7492752 = r7492751 + r7492734;
        double r7492753 = r7492741 ? r7492750 : r7492752;
        double r7492754 = r7492730 ? r7492739 : r7492753;
        return r7492754;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.2
Target15.1
Herbie0.6
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Split input into 3 regimes

  2. if eps < -65.72641980047264

    1. Initial program 29.2

      \[\sin \left(x + \varepsilon\right) - \sin x\]
    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 -65.72641980047264 < eps < 9.625760540766168e-09

    1. Initial program 44.6

      \[\sin \left(x + \varepsilon\right) - \sin x\]
    2. Using strategy rm

    3. Applied diff-sin44.6

      \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)}\]

    4. Simplified0.7

      \[\leadsto 2 \cdot \color{blue}{\left(\cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)}\]

    if 9.625760540766168e-09 < eps

    1. Initial program 30.7

      \[\sin \left(x + \varepsilon\right) - \sin x\]
    2. Using strategy rm

    3. Applied sin-sum0.5

      \[\leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x\]

    4. Applied associate--l+0.6

      \[\leadsto \color{blue}{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -65.72641980047264:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 9.625760540766168 \cdot 10^{-09}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\ \end{array}\]

Reproduce

herbie shell --seed 2019120 
(FPCore (x eps)
  :name "2sin (example 3.3)"

  :herbie-target
  (* 2 (* (cos (+ x (/ eps 2))) (sin (/ eps 2))))

  (- (sin (+ x eps)) (sin x)))