Average Error: 39.7 → 0.7
Time: 11.4s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -4.40285186539907399 \cdot 10^{-5}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 2.68480080325460108 \cdot 10^{-5}:\\ \;\;\;\;\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot -2\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -4.40285186539907399 \cdot 10^{-5}:\\
\;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\

\mathbf{elif}\;\varepsilon \le 2.68480080325460108 \cdot 10^{-5}:\\
\;\;\;\;\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot -2\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\\

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

\end{array}
double f(double x, double eps) {
        double r49831 = x;
        double r49832 = eps;
        double r49833 = r49831 + r49832;
        double r49834 = cos(r49833);
        double r49835 = cos(r49831);
        double r49836 = r49834 - r49835;
        return r49836;
}
↓
double f(double x, double eps) {
        double r49837 = eps;
        double r49838 = -4.402851865399074e-05;
        bool r49839 = r49837 <= r49838;
        double r49840 = x;
        double r49841 = cos(r49840);
        double r49842 = cos(r49837);
        double r49843 = r49841 * r49842;
        double r49844 = sin(r49840);
        double r49845 = sin(r49837);
        double r49846 = r49844 * r49845;
        double r49847 = r49843 - r49846;
        double r49848 = r49847 - r49841;
        double r49849 = 2.684800803254601e-05;
        bool r49850 = r49837 <= r49849;
        double r49851 = 0.5;
        double r49852 = r49851 * r49837;
        double r49853 = sin(r49852);
        double r49854 = -2.0;
        double r49855 = r49853 * r49854;
        double r49856 = r49840 + r49837;
        double r49857 = r49856 + r49840;
        double r49858 = 2.0;
        double r49859 = r49857 / r49858;
        double r49860 = sin(r49859);
        double r49861 = r49855 * r49860;
        double r49862 = fma(r49844, r49845, r49841);
        double r49863 = r49843 - r49862;
        double r49864 = r49850 ? r49861 : r49863;
        double r49865 = r49839 ? r49848 : r49864;
        return r49865;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 3 regimes

  2. if eps < -4.402851865399074e-05

    1. Initial program 30.2

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

    3. Applied cos-sum0.9

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

    if -4.402851865399074e-05 < eps < 2.684800803254601e-05

    1. Initial program 49.0

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

    3. Applied diff-cos37.5

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

    4. Simplified0.4

      \[\leadsto -2 \cdot \color{blue}{\left(\sin \left(\frac{0 + \varepsilon}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)}\]
    5. Using strategy rm

    6. Applied associate-*r*0.4

      \[\leadsto \color{blue}{\left(-2 \cdot \sin \left(\frac{0 + \varepsilon}{2}\right)\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)}\]

    7. Simplified0.4

      \[\leadsto \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot -2\right)} \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\]

    if 2.684800803254601e-05 < eps

    1. Initial program 31.1

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

    3. Applied cos-sum1.0

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

    4. Applied associate--l-1.0

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

    5. Simplified1.0

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -4.40285186539907399 \cdot 10^{-5}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 2.68480080325460108 \cdot 10^{-5}:\\ \;\;\;\;\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot -2\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))