Average Error: 39.8 → 0.7
Time: 15.7s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -7.45592842694085654 \cdot 10^{-5} \lor \neg \left(\varepsilon \le 9.949689884521833 \cdot 10^{-6}\right):\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\frac{\mathsf{fma}\left(x, 2, \varepsilon\right)}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\right)\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -7.45592842694085654 \cdot 10^{-5} \lor \neg \left(\varepsilon \le 9.949689884521833 \cdot 10^{-6}\right):\\
\;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\

\mathbf{else}:\\
\;\;\;\;-2 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\frac{\mathsf{fma}\left(x, 2, \varepsilon\right)}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\right)\\

\end{array}
double f(double x, double eps) {
        double r66699 = x;
        double r66700 = eps;
        double r66701 = r66699 + r66700;
        double r66702 = cos(r66701);
        double r66703 = cos(r66699);
        double r66704 = r66702 - r66703;
        return r66704;
}
↓
double f(double x, double eps) {
        double r66705 = eps;
        double r66706 = -7.455928426940857e-05;
        bool r66707 = r66705 <= r66706;
        double r66708 = 9.949689884521833e-06;
        bool r66709 = r66705 <= r66708;
        double r66710 = !r66709;
        bool r66711 = r66707 || r66710;
        double r66712 = x;
        double r66713 = cos(r66712);
        double r66714 = cos(r66705);
        double r66715 = r66713 * r66714;
        double r66716 = sin(r66712);
        double r66717 = sin(r66705);
        double r66718 = r66716 * r66717;
        double r66719 = r66715 - r66718;
        double r66720 = r66719 - r66713;
        double r66721 = -2.0;
        double r66722 = 2.0;
        double r66723 = fma(r66712, r66722, r66705);
        double r66724 = r66723 / r66722;
        double r66725 = sin(r66724);
        double r66726 = r66705 / r66722;
        double r66727 = sin(r66726);
        double r66728 = r66725 * r66727;
        double r66729 = log1p(r66728);
        double r66730 = expm1(r66729);
        double r66731 = r66721 * r66730;
        double r66732 = r66711 ? r66720 : r66731;
        return r66732;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if eps < -7.455928426940857e-05 or 9.949689884521833e-06 < eps

    1. Initial program 30.3

      \[\cos \left(x + \varepsilon\right) - \cos x\]
    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 -7.455928426940857e-05 < eps < 9.949689884521833e-06

    1. Initial program 49.6

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

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

    6. Applied expm1-log1p-u0.5

      \[\leadsto -2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\frac{\varepsilon - 0}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\right)}\]

    7. Simplified0.5

      \[\leadsto -2 \cdot \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(\sin \left(\frac{\mathsf{fma}\left(x, 2, \varepsilon\right)}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)}\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -7.45592842694085654 \cdot 10^{-5} \lor \neg \left(\varepsilon \le 9.949689884521833 \cdot 10^{-6}\right):\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\frac{\mathsf{fma}\left(x, 2, \varepsilon\right)}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))