Average Error: 37.1 → 0.8
Time: 22.6s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -7.141151458641036 \cdot 10^{-09}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 2.1464224965526296 \cdot 10^{-27}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\ \end{array}\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -7.141151458641036 \cdot 10^{-09}:\\
\;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\

\mathbf{elif}\;\varepsilon \le 2.1464224965526296 \cdot 10^{-27}:\\
\;\;\;\;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(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\

\end{array}
double f(double x, double eps) {
        double r4579727 = x;
        double r4579728 = eps;
        double r4579729 = r4579727 + r4579728;
        double r4579730 = sin(r4579729);
        double r4579731 = sin(r4579727);
        double r4579732 = r4579730 - r4579731;
        return r4579732;
}
↓
double f(double x, double eps) {
        double r4579733 = eps;
        double r4579734 = -7.141151458641036e-09;
        bool r4579735 = r4579733 <= r4579734;
        double r4579736 = x;
        double r4579737 = sin(r4579736);
        double r4579738 = cos(r4579733);
        double r4579739 = r4579737 * r4579738;
        double r4579740 = cos(r4579736);
        double r4579741 = sin(r4579733);
        double r4579742 = r4579740 * r4579741;
        double r4579743 = r4579739 + r4579742;
        double r4579744 = r4579743 - r4579737;
        double r4579745 = 2.1464224965526296e-27;
        bool r4579746 = r4579733 <= r4579745;
        double r4579747 = 2.0;
        double r4579748 = 0.5;
        double r4579749 = r4579748 * r4579733;
        double r4579750 = sin(r4579749);
        double r4579751 = r4579736 + r4579733;
        double r4579752 = r4579751 + r4579736;
        double r4579753 = r4579752 / r4579747;
        double r4579754 = cos(r4579753);
        double r4579755 = r4579750 * r4579754;
        double r4579756 = r4579747 * r4579755;
        double r4579757 = r4579742 - r4579737;
        double r4579758 = r4579757 + r4579739;
        double r4579759 = r4579746 ? r4579756 : r4579758;
        double r4579760 = r4579735 ? r4579744 : r4579759;
        return r4579760;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.1
Target14.4
Herbie0.8
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Split input into 3 regimes

  2. if eps < -7.141151458641036e-09

    1. Initial program 29.9

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

    3. Applied sin-sum0.6

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

    if -7.141151458641036e-09 < eps < 2.1464224965526296e-27

    1. Initial program 45.0

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

    3. Applied diff-sin45.0

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

    4. Simplified0.2

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

    if 2.1464224965526296e-27 < eps

    1. Initial program 29.3

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

    3. Applied sin-sum2.0

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

    4. Applied associate--l+2.0

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -7.141151458641036 \cdot 10^{-09}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 2.1464224965526296 \cdot 10^{-27}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\ \end{array}\]

Reproduce

herbie shell --seed 2019164 
(FPCore (x eps)
  :name "2sin (example 3.3)"

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

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