Average Error: 40.0 → 16.6
Time: 6.1s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.821618172153552 \cdot 10^{-16} \lor \neg \left(\varepsilon \le 4.6754847779555244 \cdot 10^{-8}\right):\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, -\frac{\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right) \cdot \left(\sin x \cdot \sin \varepsilon - \cos x\right)}{\sin x \cdot \sin \varepsilon - \cos x}\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -1.821618172153552 \cdot 10^{-16} \lor \neg \left(\varepsilon \le 4.6754847779555244 \cdot 10^{-8}\right):\\
\;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, -\frac{\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right) \cdot \left(\sin x \cdot \sin \varepsilon - \cos x\right)}{\sin x \cdot \sin \varepsilon - \cos x}\right)\\

\mathbf{else}:\\
\;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\

\end{array}
double f(double x, double eps) {
        double r54213 = x;
        double r54214 = eps;
        double r54215 = r54213 + r54214;
        double r54216 = cos(r54215);
        double r54217 = cos(r54213);
        double r54218 = r54216 - r54217;
        return r54218;
}
↓
double f(double x, double eps) {
        double r54219 = eps;
        double r54220 = -1.8216181721535518e-16;
        bool r54221 = r54219 <= r54220;
        double r54222 = 4.6754847779555244e-08;
        bool r54223 = r54219 <= r54222;
        double r54224 = !r54223;
        bool r54225 = r54221 || r54224;
        double r54226 = x;
        double r54227 = cos(r54226);
        double r54228 = cos(r54219);
        double r54229 = sin(r54226);
        double r54230 = sin(r54219);
        double r54231 = fma(r54229, r54230, r54227);
        double r54232 = r54229 * r54230;
        double r54233 = r54232 - r54227;
        double r54234 = r54231 * r54233;
        double r54235 = r54234 / r54233;
        double r54236 = -r54235;
        double r54237 = fma(r54227, r54228, r54236);
        double r54238 = 0.16666666666666666;
        double r54239 = 3.0;
        double r54240 = pow(r54226, r54239);
        double r54241 = r54238 * r54240;
        double r54242 = r54241 - r54226;
        double r54243 = 0.5;
        double r54244 = r54219 * r54243;
        double r54245 = r54242 - r54244;
        double r54246 = r54219 * r54245;
        double r54247 = r54225 ? r54237 : r54246;
        return r54247;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if eps < -1.8216181721535518e-16 or 4.6754847779555244e-08 < eps

    1. Initial program 31.2

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

    3. Applied cos-sum1.9

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

    4. Applied associate--l-1.9

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

    5. Simplified1.9

      \[\leadsto \cos x \cdot \cos \varepsilon - \color{blue}{\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)}\]
    6. Using strategy rm

    7. Applied fma-neg1.8

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos x, \cos \varepsilon, -\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\right)}\]
    8. Using strategy rm

    9. Applied fma-udef1.9

      \[\leadsto \mathsf{fma}\left(\cos x, \cos \varepsilon, -\color{blue}{\left(\sin x \cdot \sin \varepsilon + \cos x\right)}\right)\]
    10. Using strategy rm

    11. Applied flip-+2.0

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

    12. Simplified1.9

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

    if -1.8216181721535518e-16 < eps < 4.6754847779555244e-08

    1. Initial program 49.7

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

    2. Taylor expanded around 0 32.9

      \[\leadsto \color{blue}{\frac{1}{6} \cdot \left({x}^{3} \cdot \varepsilon\right) - \left(x \cdot \varepsilon + \frac{1}{2} \cdot {\varepsilon}^{2}\right)}\]

    3. Simplified32.9

      \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification16.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.821618172153552 \cdot 10^{-16} \lor \neg \left(\varepsilon \le 4.6754847779555244 \cdot 10^{-8}\right):\\ \;\;\;\;\mathsf{fma}\left(\cos x, \cos \varepsilon, -\frac{\mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right) \cdot \left(\sin x \cdot \sin \varepsilon - \cos x\right)}{\sin x \cdot \sin \varepsilon - \cos x}\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))