Average Error: 36.8 → 0.4
Time: 17.5s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -9.760584355965868752318115308952428677003 \cdot 10^{-5}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 9.433852813731427826173936825124993754699 \cdot 10^{-9}:\\ \;\;\;\;\left(2 \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\\ \end{array}\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -9.760584355965868752318115308952428677003 \cdot 10^{-5}:\\
\;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\

\mathbf{elif}\;\varepsilon \le 9.433852813731427826173936825124993754699 \cdot 10^{-9}:\\
\;\;\;\;\left(2 \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\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 r98981 = x;
        double r98982 = eps;
        double r98983 = r98981 + r98982;
        double r98984 = sin(r98983);
        double r98985 = sin(r98981);
        double r98986 = r98984 - r98985;
        return r98986;
}
↓
double f(double x, double eps) {
        double r98987 = eps;
        double r98988 = -9.760584355965869e-05;
        bool r98989 = r98987 <= r98988;
        double r98990 = x;
        double r98991 = sin(r98990);
        double r98992 = cos(r98987);
        double r98993 = r98991 * r98992;
        double r98994 = cos(r98990);
        double r98995 = sin(r98987);
        double r98996 = r98994 * r98995;
        double r98997 = r98993 + r98996;
        double r98998 = r98997 - r98991;
        double r98999 = 9.433852813731428e-09;
        bool r99000 = r98987 <= r98999;
        double r99001 = 2.0;
        double r99002 = r98987 / r99001;
        double r99003 = sin(r99002);
        double r99004 = r99001 * r99003;
        double r99005 = r98990 + r98987;
        double r99006 = r99005 + r98990;
        double r99007 = r99006 / r99001;
        double r99008 = cos(r99007);
        double r99009 = r99004 * r99008;
        double r99010 = r98996 - r98991;
        double r99011 = r98993 + r99010;
        double r99012 = r99000 ? r99009 : r99011;
        double r99013 = r98989 ? r98998 : r99012;
        return r99013;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.8
Target14.8
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 3 regimes

  2. if eps < -9.760584355965869e-05

    1. Initial program 30.4

      \[\sin \left(x + \varepsilon\right) - \sin x\]
    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 -9.760584355965869e-05 < eps < 9.433852813731428e-09

    1. Initial program 44.6

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

    3. Applied diff-sin44.6

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

    4. Simplified0.3

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

    6. Applied associate-*r*0.3

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

    if 9.433852813731428e-09 < eps

    1. Initial program 28.2

      \[\sin \left(x + \varepsilon\right) - \sin x\]
    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 simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -9.760584355965868752318115308952428677003 \cdot 10^{-5}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 9.433852813731427826173936825124993754699 \cdot 10^{-9}:\\ \;\;\;\;\left(2 \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 
(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)))