Average Error: 37.1 → 0.5
Time: 24.2s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -9.642672794131257633584719535235763032688 \cdot 10^{-5}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 9.496762876577124763011742607579646069382 \cdot 10^{-9}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\\ \end{array}\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -9.642672794131257633584719535235763032688 \cdot 10^{-5}:\\
\;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\

\mathbf{elif}\;\varepsilon \le 9.496762876577124763011742607579646069382 \cdot 10^{-9}:\\
\;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\

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

\end{array}
double f(double x, double eps) {
        double r57730 = x;
        double r57731 = eps;
        double r57732 = r57730 + r57731;
        double r57733 = sin(r57732);
        double r57734 = sin(r57730);
        double r57735 = r57733 - r57734;
        return r57735;
}
↓
double f(double x, double eps) {
        double r57736 = eps;
        double r57737 = -9.642672794131258e-05;
        bool r57738 = r57736 <= r57737;
        double r57739 = x;
        double r57740 = sin(r57739);
        double r57741 = cos(r57736);
        double r57742 = r57740 * r57741;
        double r57743 = cos(r57739);
        double r57744 = sin(r57736);
        double r57745 = r57743 * r57744;
        double r57746 = r57742 + r57745;
        double r57747 = r57746 - r57740;
        double r57748 = 9.496762876577125e-09;
        bool r57749 = r57736 <= r57748;
        double r57750 = 2.0;
        double r57751 = r57736 / r57750;
        double r57752 = sin(r57751);
        double r57753 = r57739 + r57736;
        double r57754 = r57753 + r57739;
        double r57755 = r57754 / r57750;
        double r57756 = cos(r57755);
        double r57757 = r57752 * r57756;
        double r57758 = r57750 * r57757;
        double r57759 = r57745 - r57740;
        double r57760 = r57742 + r57759;
        double r57761 = r57749 ? r57758 : r57760;
        double r57762 = r57738 ? r57747 : r57761;
        return r57762;
}

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
Target15.0
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 < -9.642672794131258e-05

    1. Initial program 30.2

      \[\sin \left(x + \varepsilon\right) - \sin x\]
    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 -9.642672794131258e-05 < eps < 9.496762876577125e-09

    1. Initial program 44.9

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

    3. Applied diff-sin44.9

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

    4. Simplified0.4

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

    if 9.496762876577125e-09 < eps

    1. Initial program 28.9

      \[\sin \left(x + \varepsilon\right) - \sin x\]
    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+0.5

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -9.642672794131257633584719535235763032688 \cdot 10^{-5}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 9.496762876577124763011742607579646069382 \cdot 10^{-9}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 
(FPCore (x eps)
  :name "2sin (example 3.3)"
  :precision binary64

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

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