Average Error: 37.2 → 0.5
Time: 15.3s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.106674806561576609834106951074650382338 \cdot 10^{-8}:\\ \;\;\;\;\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\\ \mathbf{elif}\;\varepsilon \le 1.744480835824209643740453456843553237121 \cdot 10^{-8}:\\ \;\;\;\;\left(\cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -1.106674806561576609834106951074650382338 \cdot 10^{-8}:\\
\;\;\;\;\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\\

\mathbf{elif}\;\varepsilon \le 1.744480835824209643740453456843553237121 \cdot 10^{-8}:\\
\;\;\;\;\left(\cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot 2\\

\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 r93807 = x;
        double r93808 = eps;
        double r93809 = r93807 + r93808;
        double r93810 = sin(r93809);
        double r93811 = sin(r93807);
        double r93812 = r93810 - r93811;
        return r93812;
}
↓
double f(double x, double eps) {
        double r93813 = eps;
        double r93814 = -1.1066748065615766e-08;
        bool r93815 = r93813 <= r93814;
        double r93816 = x;
        double r93817 = sin(r93816);
        double r93818 = cos(r93813);
        double r93819 = r93817 * r93818;
        double r93820 = cos(r93816);
        double r93821 = sin(r93813);
        double r93822 = r93820 * r93821;
        double r93823 = r93822 - r93817;
        double r93824 = r93819 + r93823;
        double r93825 = 1.7444808358242096e-08;
        bool r93826 = r93813 <= r93825;
        double r93827 = r93813 + r93816;
        double r93828 = r93816 + r93827;
        double r93829 = 2.0;
        double r93830 = r93828 / r93829;
        double r93831 = cos(r93830);
        double r93832 = r93813 / r93829;
        double r93833 = sin(r93832);
        double r93834 = r93831 * r93833;
        double r93835 = r93834 * r93829;
        double r93836 = r93819 + r93822;
        double r93837 = r93836 - r93817;
        double r93838 = r93826 ? r93835 : r93837;
        double r93839 = r93815 ? r93824 : r93838;
        return r93839;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.2
Target15.5
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 < -1.1066748065615766e-08

    1. Initial program 30.3

      \[\sin \left(x + \varepsilon\right) - \sin x\]
    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+0.7

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

    if -1.1066748065615766e-08 < eps < 1.7444808358242096e-08

    1. Initial program 44.6

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

    3. Applied diff-sin44.7

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

    4. Simplified0.3

      \[\leadsto 2 \cdot \color{blue}{\left(\sin \left(\frac{\varepsilon + 0}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)}\]
    5. Using strategy rm

    6. Applied associate-+l+0.3

      \[\leadsto 2 \cdot \left(\sin \left(\frac{\varepsilon + 0}{2}\right) \cdot \cos \left(\frac{\color{blue}{x + \left(\varepsilon + x\right)}}{2}\right)\right)\]

    if 1.7444808358242096e-08 < eps

    1. Initial program 30.2

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

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.106674806561576609834106951074650382338 \cdot 10^{-8}:\\ \;\;\;\;\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\\ \mathbf{elif}\;\varepsilon \le 1.744480835824209643740453456843553237121 \cdot 10^{-8}:\\ \;\;\;\;\left(\cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]

Reproduce

herbie shell --seed 2019209 +o rules:numerics
(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)))