Average Error: 39.6 → 0.8
Time: 36.7s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -2.0140999975557835 \cdot 10^{-07}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 0.00017141206035122596:\\ \;\;\;\;\mathsf{log1p}\left(\left(\mathsf{expm1}\left(\left(\sin \left(\frac{x + \left(x + \varepsilon\right)}{2}\right) \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot -2\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \mathsf{fma}\left(\left(\sin \varepsilon\right), \left(\sin x\right), \left(\cos x\right)\right)\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -2.0140999975557835 \cdot 10^{-07}:\\
\;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\

\mathbf{elif}\;\varepsilon \le 0.00017141206035122596:\\
\;\;\;\;\mathsf{log1p}\left(\left(\mathsf{expm1}\left(\left(\sin \left(\frac{x + \left(x + \varepsilon\right)}{2}\right) \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot -2\right)\right)\right)\right)\right)\\

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

\end{array}
double f(double x, double eps) {
        double r4234853 = x;
        double r4234854 = eps;
        double r4234855 = r4234853 + r4234854;
        double r4234856 = cos(r4234855);
        double r4234857 = cos(r4234853);
        double r4234858 = r4234856 - r4234857;
        return r4234858;
}
↓
double f(double x, double eps) {
        double r4234859 = eps;
        double r4234860 = -2.0140999975557835e-07;
        bool r4234861 = r4234859 <= r4234860;
        double r4234862 = x;
        double r4234863 = cos(r4234862);
        double r4234864 = cos(r4234859);
        double r4234865 = r4234863 * r4234864;
        double r4234866 = sin(r4234862);
        double r4234867 = sin(r4234859);
        double r4234868 = r4234866 * r4234867;
        double r4234869 = r4234865 - r4234868;
        double r4234870 = r4234869 - r4234863;
        double r4234871 = 0.00017141206035122596;
        bool r4234872 = r4234859 <= r4234871;
        double r4234873 = r4234862 + r4234859;
        double r4234874 = r4234862 + r4234873;
        double r4234875 = 2.0;
        double r4234876 = r4234874 / r4234875;
        double r4234877 = sin(r4234876);
        double r4234878 = r4234859 / r4234875;
        double r4234879 = sin(r4234878);
        double r4234880 = -2.0;
        double r4234881 = r4234879 * r4234880;
        double r4234882 = r4234877 * r4234881;
        double r4234883 = expm1(r4234882);
        double r4234884 = log1p(r4234883);
        double r4234885 = fma(r4234867, r4234866, r4234863);
        double r4234886 = r4234865 - r4234885;
        double r4234887 = r4234872 ? r4234884 : r4234886;
        double r4234888 = r4234861 ? r4234870 : r4234887;
        return r4234888;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 3 regimes

  2. if eps < -2.0140999975557835e-07

    1. Initial program 30.1

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

    3. Applied cos-sum1.1

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

    if -2.0140999975557835e-07 < eps < 0.00017141206035122596

    1. Initial program 48.9

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

    3. Applied diff-cos37.4

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

    4. Simplified0.5

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

    6. Applied associate-*r*0.5

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

    8. Applied expm1-log1p-u0.6

      \[\leadsto \left(-2 \cdot \color{blue}{\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left(\sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)\right)\right)\right)\right)}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\]
    9. Using strategy rm

    10. Applied log1p-expm1-u0.6

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\left(\mathsf{expm1}\left(\left(\left(-2 \cdot \mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left(\sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)\right)\right)\right)\right)\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\right)\right)\right)}\]

    11. Simplified0.5

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\left(\mathsf{expm1}\left(\left(\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot -2\right) \cdot \sin \left(\frac{x + \left(x + \varepsilon\right)}{2}\right)\right)\right)\right)}\right)\]

    if 0.00017141206035122596 < eps

    1. Initial program 30.1

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

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

    5. Simplified0.9

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -2.0140999975557835 \cdot 10^{-07}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 0.00017141206035122596:\\ \;\;\;\;\mathsf{log1p}\left(\left(\mathsf{expm1}\left(\left(\sin \left(\frac{x + \left(x + \varepsilon\right)}{2}\right) \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot -2\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \mathsf{fma}\left(\left(\sin \varepsilon\right), \left(\sin x\right), \left(\cos x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019125 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))