Average Error: 39.8 → 0.7
Time: 14.5s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -7.45592842694085654 \cdot 10^{-5} \lor \neg \left(\varepsilon \le 9.949689884521833 \cdot 10^{-6}\right):\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\right)\\ \end{array}\]
\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(2, x, \varepsilon\right)}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\right)\\

\end{array}
double f(double x, double eps) {
        double r70028 = x;
        double r70029 = eps;
        double r70030 = r70028 + r70029;
        double r70031 = cos(r70030);
        double r70032 = cos(r70028);
        double r70033 = r70031 - r70032;
        return r70033;
}
↓
double f(double x, double eps) {
        double r70034 = eps;
        double r70035 = -7.455928426940857e-05;
        bool r70036 = r70034 <= r70035;
        double r70037 = 9.949689884521833e-06;
        bool r70038 = r70034 <= r70037;
        double r70039 = !r70038;
        bool r70040 = r70036 || r70039;
        double r70041 = x;
        double r70042 = cos(r70041);
        double r70043 = cos(r70034);
        double r70044 = r70042 * r70043;
        double r70045 = sin(r70041);
        double r70046 = sin(r70034);
        double r70047 = r70045 * r70046;
        double r70048 = r70044 - r70047;
        double r70049 = r70048 - r70042;
        double r70050 = -2.0;
        double r70051 = 2.0;
        double r70052 = fma(r70051, r70041, r70034);
        double r70053 = r70052 / r70051;
        double r70054 = sin(r70053);
        double r70055 = r70034 / r70051;
        double r70056 = sin(r70055);
        double r70057 = r70054 * r70056;
        double r70058 = log1p(r70057);
        double r70059 = expm1(r70058);
        double r70060 = r70050 * r70059;
        double r70061 = r70040 ? r70049 : r70060;
        return r70061;
}

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

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

  4. Final simplification0.7

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

Reproduce

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