Average Error: 37.0 → 0.5
Time: 15.8s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -7.672969354734922 \cdot 10^{-09}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 1.4909178493851906 \cdot 10^{-15}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -7.672969354734922 \cdot 10^{-09}:\\
\;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\

\mathbf{elif}\;\varepsilon \le 1.4909178493851906 \cdot 10^{-15}:\\
\;\;\;\;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 r2216781 = x;
        double r2216782 = eps;
        double r2216783 = r2216781 + r2216782;
        double r2216784 = sin(r2216783);
        double r2216785 = sin(r2216781);
        double r2216786 = r2216784 - r2216785;
        return r2216786;
}
↓
double f(double x, double eps) {
        double r2216787 = eps;
        double r2216788 = -7.672969354734922e-09;
        bool r2216789 = r2216787 <= r2216788;
        double r2216790 = x;
        double r2216791 = sin(r2216790);
        double r2216792 = cos(r2216787);
        double r2216793 = r2216791 * r2216792;
        double r2216794 = cos(r2216790);
        double r2216795 = sin(r2216787);
        double r2216796 = r2216794 * r2216795;
        double r2216797 = r2216793 + r2216796;
        double r2216798 = r2216797 - r2216791;
        double r2216799 = 1.4909178493851906e-15;
        bool r2216800 = r2216787 <= r2216799;
        double r2216801 = 2.0;
        double r2216802 = 0.5;
        double r2216803 = r2216802 * r2216787;
        double r2216804 = sin(r2216803);
        double r2216805 = r2216790 + r2216787;
        double r2216806 = r2216805 + r2216790;
        double r2216807 = r2216806 / r2216801;
        double r2216808 = cos(r2216807);
        double r2216809 = r2216804 * r2216808;
        double r2216810 = r2216801 * r2216809;
        double r2216811 = r2216800 ? r2216810 : r2216798;
        double r2216812 = r2216789 ? r2216798 : r2216811;
        return r2216812;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.0
Target14.7
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 2 regimes

  2. if eps < -7.672969354734922e-09 or 1.4909178493851906e-15 < eps

    1. Initial program 29.4

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

    3. Applied sin-sum0.7

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

    if -7.672969354734922e-09 < eps < 1.4909178493851906e-15

    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. Simplified0.2

      \[\leadsto 2 \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos \left(\frac{x + \left(x + \varepsilon\right)}{2}\right)\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -7.672969354734922 \cdot 10^{-09}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 1.4909178493851906 \cdot 10^{-15}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]

Reproduce

herbie shell --seed 2019154 
(FPCore (x eps)
  :name "2sin (example 3.3)"

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

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