Average Error: 37.3 → 0.4
Time: 15.7s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.379308090176633415808603902149090392193 \cdot 10^{-8}:\\ \;\;\;\;\left(\cos \varepsilon \cdot \sin x + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 6.52085484414657072417395637610113001692 \cdot 10^{-9}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \cos \varepsilon \cdot \sin x\\ \end{array}\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -1.379308090176633415808603902149090392193 \cdot 10^{-8}:\\
\;\;\;\;\left(\cos \varepsilon \cdot \sin x + \cos x \cdot \sin \varepsilon\right) - \sin x\\

\mathbf{elif}\;\varepsilon \le 6.52085484414657072417395637610113001692 \cdot 10^{-9}:\\
\;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)\right)\\

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

\end{array}
double f(double x, double eps) {
        double r138844 = x;
        double r138845 = eps;
        double r138846 = r138844 + r138845;
        double r138847 = sin(r138846);
        double r138848 = sin(r138844);
        double r138849 = r138847 - r138848;
        return r138849;
}
↓
double f(double x, double eps) {
        double r138850 = eps;
        double r138851 = -1.3793080901766334e-08;
        bool r138852 = r138850 <= r138851;
        double r138853 = cos(r138850);
        double r138854 = x;
        double r138855 = sin(r138854);
        double r138856 = r138853 * r138855;
        double r138857 = cos(r138854);
        double r138858 = sin(r138850);
        double r138859 = r138857 * r138858;
        double r138860 = r138856 + r138859;
        double r138861 = r138860 - r138855;
        double r138862 = 6.520854844146571e-09;
        bool r138863 = r138850 <= r138862;
        double r138864 = 2.0;
        double r138865 = r138850 / r138864;
        double r138866 = sin(r138865);
        double r138867 = r138850 + r138854;
        double r138868 = r138854 + r138867;
        double r138869 = r138868 / r138864;
        double r138870 = cos(r138869);
        double r138871 = r138866 * r138870;
        double r138872 = r138864 * r138871;
        double r138873 = r138859 - r138855;
        double r138874 = r138873 + r138856;
        double r138875 = r138863 ? r138872 : r138874;
        double r138876 = r138852 ? r138861 : r138875;
        return r138876;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.3
Target14.9
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 < -1.3793080901766334e-08

    1. Initial program 29.7

      \[\sin \left(x + \varepsilon\right) - \sin x\]
    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. Simplified0.5

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

    if -1.3793080901766334e-08 < eps < 6.520854844146571e-09

    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

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

    if 6.520854844146571e-09 < eps

    1. Initial program 30.1

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

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.379308090176633415808603902149090392193 \cdot 10^{-8}:\\ \;\;\;\;\left(\cos \varepsilon \cdot \sin x + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 6.52085484414657072417395637610113001692 \cdot 10^{-9}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \cos \varepsilon \cdot \sin x\\ \end{array}\]

Reproduce

herbie shell --seed 2019196 
(FPCore (x eps)
  :name "2sin (example 3.3)"

  :herbie-target
  (* 2.0 (* (cos (+ x (/ eps 2.0))) (sin (/ eps 2.0))))

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