Average Error: 36.9 → 0.6
Time: 17.1s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -6443.306290177518349082674831151962280273:\\ \;\;\;\;\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\\ \mathbf{elif}\;\varepsilon \le 5.463141894552383288993626720472102098469 \cdot 10^{-9}:\\ \;\;\;\;2 \cdot \left(\cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\\ \end{array}\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -6443.306290177518349082674831151962280273:\\
\;\;\;\;\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\\

\mathbf{elif}\;\varepsilon \le 5.463141894552383288993626720472102098469 \cdot 10^{-9}:\\
\;\;\;\;2 \cdot \left(\cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right) \cdot \sin \left(\frac{\varepsilon}{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 r6352783 = x;
        double r6352784 = eps;
        double r6352785 = r6352783 + r6352784;
        double r6352786 = sin(r6352785);
        double r6352787 = sin(r6352783);
        double r6352788 = r6352786 - r6352787;
        return r6352788;
}
↓
double f(double x, double eps) {
        double r6352789 = eps;
        double r6352790 = -6443.306290177518;
        bool r6352791 = r6352789 <= r6352790;
        double r6352792 = x;
        double r6352793 = sin(r6352792);
        double r6352794 = cos(r6352789);
        double r6352795 = r6352793 * r6352794;
        double r6352796 = cos(r6352792);
        double r6352797 = sin(r6352789);
        double r6352798 = r6352796 * r6352797;
        double r6352799 = r6352798 - r6352793;
        double r6352800 = r6352795 + r6352799;
        double r6352801 = 5.463141894552383e-09;
        bool r6352802 = r6352789 <= r6352801;
        double r6352803 = 2.0;
        double r6352804 = r6352792 + r6352789;
        double r6352805 = r6352804 + r6352792;
        double r6352806 = r6352805 / r6352803;
        double r6352807 = cos(r6352806);
        double r6352808 = r6352789 / r6352803;
        double r6352809 = sin(r6352808);
        double r6352810 = r6352807 * r6352809;
        double r6352811 = r6352803 * r6352810;
        double r6352812 = r6352802 ? r6352811 : r6352800;
        double r6352813 = r6352791 ? r6352800 : r6352812;
        return r6352813;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.9
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 2 regimes

  2. if eps < -6443.306290177518 or 5.463141894552383e-09 < eps

    1. Initial program 29.9

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

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

    if -6443.306290177518 < eps < 5.463141894552383e-09

    1. Initial program 44.2

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

    3. Applied diff-sin44.2

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

    4. Simplified0.7

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -6443.306290177518349082674831151962280273:\\ \;\;\;\;\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\\ \mathbf{elif}\;\varepsilon \le 5.463141894552383288993626720472102098469 \cdot 10^{-9}:\\ \;\;\;\;2 \cdot \left(\cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(FPCore (x eps)
  :name "2sin (example 3.3)"

  :herbie-target
  (* 2.0 (* (cos (+ x (/ eps 2.0))) (sin (/ eps 2.0))))

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