Average Error: 36.8 → 0.4
Time: 23.8s
Precision: 64
\[\sin \left(x + \varepsilon\right) - \sin x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -9.108842191445869 \cdot 10^{-09}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 1.2412047115655244 \cdot 10^{-08}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{x + \left(x + \varepsilon\right)}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\ \end{array}\]
\sin \left(x + \varepsilon\right) - \sin x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -9.108842191445869 \cdot 10^{-09}:\\
\;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\

\mathbf{elif}\;\varepsilon \le 1.2412047115655244 \cdot 10^{-08}:\\
\;\;\;\;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 r3225883 = x;
        double r3225884 = eps;
        double r3225885 = r3225883 + r3225884;
        double r3225886 = sin(r3225885);
        double r3225887 = sin(r3225883);
        double r3225888 = r3225886 - r3225887;
        return r3225888;
}
↓
double f(double x, double eps) {
        double r3225889 = eps;
        double r3225890 = -9.108842191445869e-09;
        bool r3225891 = r3225889 <= r3225890;
        double r3225892 = x;
        double r3225893 = sin(r3225892);
        double r3225894 = cos(r3225889);
        double r3225895 = r3225893 * r3225894;
        double r3225896 = cos(r3225892);
        double r3225897 = sin(r3225889);
        double r3225898 = r3225896 * r3225897;
        double r3225899 = r3225895 + r3225898;
        double r3225900 = r3225899 - r3225893;
        double r3225901 = 1.2412047115655244e-08;
        bool r3225902 = r3225889 <= r3225901;
        double r3225903 = 2.0;
        double r3225904 = r3225889 / r3225903;
        double r3225905 = sin(r3225904);
        double r3225906 = r3225892 + r3225889;
        double r3225907 = r3225892 + r3225906;
        double r3225908 = r3225907 / r3225903;
        double r3225909 = cos(r3225908);
        double r3225910 = r3225905 * r3225909;
        double r3225911 = r3225903 * r3225910;
        double r3225912 = r3225898 - r3225893;
        double r3225913 = r3225912 + r3225895;
        double r3225914 = r3225902 ? r3225911 : r3225913;
        double r3225915 = r3225891 ? r3225900 : r3225914;
        return r3225915;
}

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
Target14.6
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 < -9.108842191445869e-09

    1. Initial program 29.1

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

    3. Applied sin-sum0.5

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

    if -9.108842191445869e-09 < eps < 1.2412047115655244e-08

    1. Initial program 45.1

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

    3. Applied diff-sin45.1

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

    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 1.2412047115655244e-08 < eps

    1. Initial program 28.7

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

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -9.108842191445869 \cdot 10^{-09}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 1.2412047115655244 \cdot 10^{-08}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{x + \left(x + \varepsilon\right)}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \sin \varepsilon - \sin x\right) + \sin x \cdot \cos \varepsilon\\ \end{array}\]

Reproduce

herbie shell --seed 2019146 
(FPCore (x eps)
  :name "2sin (example 3.3)"

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

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