Average Error: 37.5 → 0.5
Time: 16.7s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.1998709236678226 \cdot 10^{-08}:\\ \;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\ \mathbf{elif}\;\varepsilon \le 1.1087986248072222 \cdot 10^{-08}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -1.1998709236678226 \cdot 10^{-08}:\\
\;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\

\mathbf{elif}\;\varepsilon \le 1.1087986248072222 \cdot 10^{-08}:\\
\;\;\;\;2 \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\

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

\end{array}
double f(double x, double eps) {
        double r5718836 = x;
        double r5718837 = eps;
        double r5718838 = r5718836 + r5718837;
        double r5718839 = sin(r5718838);
        double r5718840 = sin(r5718836);
        double r5718841 = r5718839 - r5718840;
        return r5718841;
}
↓
double f(double x, double eps) {
        double r5718842 = eps;
        double r5718843 = -1.1998709236678226e-08;
        bool r5718844 = r5718842 <= r5718843;
        double r5718845 = x;
        double r5718846 = cos(r5718845);
        double r5718847 = sin(r5718842);
        double r5718848 = r5718846 * r5718847;
        double r5718849 = sin(r5718845);
        double r5718850 = r5718848 - r5718849;
        double r5718851 = cos(r5718842);
        double r5718852 = r5718849 * r5718851;
        double r5718853 = r5718850 + r5718852;
        double r5718854 = 1.1087986248072222e-08;
        bool r5718855 = r5718842 <= r5718854;
        double r5718856 = 2.0;
        double r5718857 = 0.5;
        double r5718858 = r5718857 * r5718842;
        double r5718859 = sin(r5718858);
        double r5718860 = r5718845 + r5718842;
        double r5718861 = r5718860 + r5718845;
        double r5718862 = r5718861 / r5718856;
        double r5718863 = cos(r5718862);
        double r5718864 = r5718859 * r5718863;
        double r5718865 = r5718856 * r5718864;
        double r5718866 = r5718852 + r5718848;
        double r5718867 = r5718866 - r5718849;
        double r5718868 = r5718855 ? r5718865 : r5718867;
        double r5718869 = r5718844 ? r5718853 : r5718868;
        return r5718869;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.5
Target15.6
Herbie0.5
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Split input into 3 regimes

  2. if eps < -1.1998709236678226e-08

    1. Initial program 31.1

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

    3. Applied sin-sum0.6

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

    4. Applied associate--l+0.6

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

    if -1.1998709236678226e-08 < eps < 1.1087986248072222e-08

    1. Initial program 44.4

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

    3. Applied diff-sin44.4

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

    4. Simplified0.3

      \[\leadsto 2 \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos \left(\frac{x + \left(x + \varepsilon\right)}{2}\right)\right)}\]

    if 1.1087986248072222e-08 < eps

    1. Initial program 30.9

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

    3. Applied sin-sum0.6

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.1998709236678226 \cdot 10^{-08}:\\ \;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\ \mathbf{elif}\;\varepsilon \le 1.1087986248072222 \cdot 10^{-08}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 
(FPCore (x eps)
  :name "2sin (example 3.3)"

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

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