Average Error: 36.6 → 0.6
Time: 24.4s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -7.894373646089193 \cdot 10^{-09}:\\ \;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\ \mathbf{elif}\;\varepsilon \le 2.1490204748102395 \cdot 10^{-17}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\ \end{array}\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -7.894373646089193 \cdot 10^{-09}:\\
\;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\

\mathbf{elif}\;\varepsilon \le 2.1490204748102395 \cdot 10^{-17}:\\
\;\;\;\;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 r17373849 = x;
        double r17373850 = eps;
        double r17373851 = r17373849 + r17373850;
        double r17373852 = sin(r17373851);
        double r17373853 = sin(r17373849);
        double r17373854 = r17373852 - r17373853;
        return r17373854;
}
↓
double f(double x, double eps) {
        double r17373855 = eps;
        double r17373856 = -7.894373646089193e-09;
        bool r17373857 = r17373855 <= r17373856;
        double r17373858 = x;
        double r17373859 = cos(r17373858);
        double r17373860 = sin(r17373855);
        double r17373861 = r17373859 * r17373860;
        double r17373862 = sin(r17373858);
        double r17373863 = r17373861 - r17373862;
        double r17373864 = cos(r17373855);
        double r17373865 = r17373862 * r17373864;
        double r17373866 = r17373863 + r17373865;
        double r17373867 = 2.1490204748102395e-17;
        bool r17373868 = r17373855 <= r17373867;
        double r17373869 = 2.0;
        double r17373870 = r17373855 / r17373869;
        double r17373871 = sin(r17373870);
        double r17373872 = r17373858 + r17373855;
        double r17373873 = r17373872 + r17373858;
        double r17373874 = r17373873 / r17373869;
        double r17373875 = cos(r17373874);
        double r17373876 = r17373871 * r17373875;
        double r17373877 = r17373869 * r17373876;
        double r17373878 = r17373868 ? r17373877 : r17373866;
        double r17373879 = r17373857 ? r17373866 : r17373878;
        return r17373879;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.6
Target15.3
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 2 regimes

  2. if eps < -7.894373646089193e-09 or 2.1490204748102395e-17 < eps

    1. Initial program 29.8

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

    3. Applied sin-sum0.8

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

    4. Applied associate--l+0.8

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

    if -7.894373646089193e-09 < eps < 2.1490204748102395e-17

    1. Initial program 44.3

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

    3. Applied diff-sin44.3

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

    4. Simplified0.2

      \[\leadsto 2 \cdot \color{blue}{\left(\cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -7.894373646089193 \cdot 10^{-09}:\\ \;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\ \mathbf{elif}\;\varepsilon \le 2.1490204748102395 \cdot 10^{-17}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\ \end{array}\]

Reproduce

herbie shell --seed 2019119 +o rules:numerics
(FPCore (x eps)
  :name "2sin (example 3.3)"

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

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