Average Error: 39.8 → 0.9
Time: 22.6s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1576820.31278729089535772800445556640625:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 2.264234444891437172519416121119206763979 \cdot 10^{-4}:\\ \;\;\;\;\left(-2 \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -1576820.31278729089535772800445556640625:\\
\;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\

\mathbf{elif}\;\varepsilon \le 2.264234444891437172519416121119206763979 \cdot 10^{-4}:\\
\;\;\;\;\left(-2 \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\\

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

\end{array}
double f(double x, double eps) {
        double r19864 = x;
        double r19865 = eps;
        double r19866 = r19864 + r19865;
        double r19867 = cos(r19866);
        double r19868 = cos(r19864);
        double r19869 = r19867 - r19868;
        return r19869;
}
↓
double f(double x, double eps) {
        double r19870 = eps;
        double r19871 = -1576820.312787291;
        bool r19872 = r19870 <= r19871;
        double r19873 = x;
        double r19874 = cos(r19873);
        double r19875 = cos(r19870);
        double r19876 = r19874 * r19875;
        double r19877 = sin(r19873);
        double r19878 = sin(r19870);
        double r19879 = r19877 * r19878;
        double r19880 = r19876 - r19879;
        double r19881 = r19880 - r19874;
        double r19882 = 0.00022642344448914372;
        bool r19883 = r19870 <= r19882;
        double r19884 = -2.0;
        double r19885 = 2.0;
        double r19886 = r19870 / r19885;
        double r19887 = sin(r19886);
        double r19888 = r19884 * r19887;
        double r19889 = r19873 + r19870;
        double r19890 = r19889 + r19873;
        double r19891 = r19890 / r19885;
        double r19892 = sin(r19891);
        double r19893 = r19888 * r19892;
        double r19894 = r19879 + r19874;
        double r19895 = r19876 - r19894;
        double r19896 = r19883 ? r19893 : r19895;
        double r19897 = r19872 ? r19881 : r19896;
        return r19897;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if eps < -1576820.312787291

    1. Initial program 30.2

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

    3. Applied cos-sum0.8

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

    if -1576820.312787291 < eps < 0.00022642344448914372

    1. Initial program 49.0

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

    3. Applied diff-cos38.0

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

    4. Simplified0.9

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

    6. Applied associate-*r*0.9

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

    if 0.00022642344448914372 < eps

    1. Initial program 31.1

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

    3. Applied cos-sum0.9

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

    4. Applied associate--l-0.9

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1576820.31278729089535772800445556640625:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 2.264234444891437172519416121119206763979 \cdot 10^{-4}:\\ \;\;\;\;\left(-2 \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))