Average Error: 36.5 → 0.6
Time: 25.1s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -0.0006966600469417059:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 8.159405127878992 \cdot 10^{-23}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{x + \left(x + \varepsilon\right)}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\ \end{array}\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -0.0006966600469417059:\\
\;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\

\mathbf{elif}\;\varepsilon \le 8.159405127878992 \cdot 10^{-23}:\\
\;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{x + \left(x + \varepsilon\right)}{2}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\

\end{array}
double f(double x, double eps) {
        double r4119955 = x;
        double r4119956 = eps;
        double r4119957 = r4119955 + r4119956;
        double r4119958 = sin(r4119957);
        double r4119959 = sin(r4119955);
        double r4119960 = r4119958 - r4119959;
        return r4119960;
}
↓
double f(double x, double eps) {
        double r4119961 = eps;
        double r4119962 = -0.0006966600469417059;
        bool r4119963 = r4119961 <= r4119962;
        double r4119964 = x;
        double r4119965 = sin(r4119964);
        double r4119966 = cos(r4119961);
        double r4119967 = r4119965 * r4119966;
        double r4119968 = cos(r4119964);
        double r4119969 = sin(r4119961);
        double r4119970 = r4119968 * r4119969;
        double r4119971 = r4119967 + r4119970;
        double r4119972 = r4119971 - r4119965;
        double r4119973 = 8.159405127878992e-23;
        bool r4119974 = r4119961 <= r4119973;
        double r4119975 = 2.0;
        double r4119976 = r4119961 / r4119975;
        double r4119977 = sin(r4119976);
        double r4119978 = r4119964 + r4119961;
        double r4119979 = r4119964 + r4119978;
        double r4119980 = r4119979 / r4119975;
        double r4119981 = cos(r4119980);
        double r4119982 = r4119977 * r4119981;
        double r4119983 = r4119975 * r4119982;
        double r4119984 = r4119970 - r4119965;
        double r4119985 = r4119984 + r4119967;
        double r4119986 = r4119974 ? r4119983 : r4119985;
        double r4119987 = r4119963 ? r4119972 : r4119986;
        return r4119987;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original36.5
Target14.8
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 3 regimes

  2. if eps < -0.0006966600469417059

    1. Initial program 28.8

      \[\sin \left(x + \varepsilon\right) - \sin x\]
    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 -0.0006966600469417059 < eps < 8.159405127878992e-23

    1. Initial program 44.3

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

    3. Applied diff-sin44.3

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

    4. Simplified0.3

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

    if 8.159405127878992e-23 < eps

    1. Initial program 29.6

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

    3. Applied sin-sum1.4

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

    4. Applied associate--l+1.4

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -0.0006966600469417059:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 8.159405127878992 \cdot 10^{-23}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{x + \left(x + \varepsilon\right)}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\ \end{array}\]

Reproduce

herbie shell --seed 2019149 
(FPCore (x eps)
  :name "2sin (example 3.3)"

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

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