Average Error: 37.1 → 0.4
Time: 22.1s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.1927174858984138 \cdot 10^{-08}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 1.1212450288248926 \cdot 10^{-08}:\\ \;\;\;\;\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(2 \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -1.1927174858984138 \cdot 10^{-08}:\\
\;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\

\mathbf{elif}\;\varepsilon \le 1.1212450288248926 \cdot 10^{-08}:\\
\;\;\;\;\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(2 \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 r7347886 = x;
        double r7347887 = eps;
        double r7347888 = r7347886 + r7347887;
        double r7347889 = sin(r7347888);
        double r7347890 = sin(r7347886);
        double r7347891 = r7347889 - r7347890;
        return r7347891;
}
↓
double f(double x, double eps) {
        double r7347892 = eps;
        double r7347893 = -1.1927174858984138e-08;
        bool r7347894 = r7347892 <= r7347893;
        double r7347895 = x;
        double r7347896 = sin(r7347895);
        double r7347897 = cos(r7347892);
        double r7347898 = r7347896 * r7347897;
        double r7347899 = cos(r7347895);
        double r7347900 = sin(r7347892);
        double r7347901 = r7347899 * r7347900;
        double r7347902 = r7347898 + r7347901;
        double r7347903 = r7347902 - r7347896;
        double r7347904 = 1.1212450288248926e-08;
        bool r7347905 = r7347892 <= r7347904;
        double r7347906 = 2.0;
        double r7347907 = r7347892 / r7347906;
        double r7347908 = sin(r7347907);
        double r7347909 = r7347895 + r7347892;
        double r7347910 = r7347909 + r7347895;
        double r7347911 = r7347910 / r7347906;
        double r7347912 = cos(r7347911);
        double r7347913 = r7347906 * r7347912;
        double r7347914 = r7347908 * r7347913;
        double r7347915 = r7347905 ? r7347914 : r7347903;
        double r7347916 = r7347894 ? r7347903 : r7347915;
        return r7347916;
}

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
Target15.1
Herbie0.4
\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

  1. Split input into 2 regimes

  2. if eps < -1.1927174858984138e-08 or 1.1212450288248926e-08 < eps

    1. Initial program 29.8

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

    3. Applied sin-sum0.5

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

    if -1.1927174858984138e-08 < eps < 1.1212450288248926e-08

    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. Simplified0.3

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

    6. Applied associate-*r*0.3

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.1927174858984138 \cdot 10^{-08}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 1.1212450288248926 \cdot 10^{-08}:\\ \;\;\;\;\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(2 \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]

Reproduce

herbie shell --seed 2019121 +o rules:numerics
(FPCore (x eps)
  :name "2sin (example 3.3)"

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

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