Average Error: 39.1 → 15.8
Time: 6.4s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.320893942837389302245904465927389681568 \cdot 10^{-9} \lor \neg \left(\varepsilon \le 9.805431973736002190838386379022166561281 \cdot 10^{-10}\right):\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \left(\sin \varepsilon - 0\right) + \cos x\right)\right) + \mathsf{fma}\left(-\cos x, 1, \cos x \cdot 1\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -1.320893942837389302245904465927389681568 \cdot 10^{-9} \lor \neg \left(\varepsilon \le 9.805431973736002190838386379022166561281 \cdot 10^{-10}\right):\\
\;\;\;\;\left(\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \left(\sin \varepsilon - 0\right) + \cos x\right)\right) + \mathsf{fma}\left(-\cos x, 1, \cos x \cdot 1\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 r49231 = x;
        double r49232 = eps;
        double r49233 = r49231 + r49232;
        double r49234 = cos(r49233);
        double r49235 = cos(r49231);
        double r49236 = r49234 - r49235;
        return r49236;
}
↓
double f(double x, double eps) {
        double r49237 = eps;
        double r49238 = -1.3208939428373893e-09;
        bool r49239 = r49237 <= r49238;
        double r49240 = 9.805431973736002e-10;
        bool r49241 = r49237 <= r49240;
        double r49242 = !r49241;
        bool r49243 = r49239 || r49242;
        double r49244 = x;
        double r49245 = cos(r49244);
        double r49246 = cos(r49237);
        double r49247 = r49245 * r49246;
        double r49248 = sin(r49244);
        double r49249 = sin(r49237);
        double r49250 = 0.0;
        double r49251 = r49249 - r49250;
        double r49252 = r49248 * r49251;
        double r49253 = r49252 + r49245;
        double r49254 = r49247 - r49253;
        double r49255 = -r49245;
        double r49256 = 1.0;
        double r49257 = r49245 * r49256;
        double r49258 = fma(r49255, r49256, r49257);
        double r49259 = r49254 + r49258;
        double r49260 = 0.16666666666666666;
        double r49261 = 3.0;
        double r49262 = pow(r49244, r49261);
        double r49263 = r49260 * r49262;
        double r49264 = r49263 - r49244;
        double r49265 = 0.5;
        double r49266 = r49237 * r49265;
        double r49267 = r49264 - r49266;
        double r49268 = r49237 * r49267;
        double r49269 = r49243 ? r49259 : r49268;
        return r49269;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if eps < -1.3208939428373893e-09 or 9.805431973736002e-10 < eps

    1. Initial program 30.1

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

    3. Applied cos-sum1.5

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

    5. Applied prod-diff1.5

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

    6. Applied associate--l+1.5

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

    7. Simplified1.5

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

    9. Applied *-un-lft-identity1.5

      \[\leadsto \mathsf{fma}\left(\cos x, \cos \varepsilon, -\sin \varepsilon \cdot \sin x\right) + \left(\sin x \cdot \left(\left(-\sin \varepsilon\right) + \sin \varepsilon\right) - \color{blue}{1 \cdot \cos x}\right)\]

    10. Applied prod-diff1.5

      \[\leadsto \mathsf{fma}\left(\cos x, \cos \varepsilon, -\sin \varepsilon \cdot \sin x\right) + \color{blue}{\left(\mathsf{fma}\left(\sin x, \left(-\sin \varepsilon\right) + \sin \varepsilon, -\cos x \cdot 1\right) + \mathsf{fma}\left(-\cos x, 1, \cos x \cdot 1\right)\right)}\]

    11. Applied associate-+r+1.5

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\cos x, \cos \varepsilon, -\sin \varepsilon \cdot \sin x\right) + \mathsf{fma}\left(\sin x, \left(-\sin \varepsilon\right) + \sin \varepsilon, -\cos x \cdot 1\right)\right) + \mathsf{fma}\left(-\cos x, 1, \cos x \cdot 1\right)}\]

    12. Simplified1.5

      \[\leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \left(\sin \varepsilon - 0\right) + \cos x\right)\right)} + \mathsf{fma}\left(-\cos x, 1, \cos x \cdot 1\right)\]

    if -1.3208939428373893e-09 < eps < 9.805431973736002e-10

    1. Initial program 48.8

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

    2. Taylor expanded around 0 31.1

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

    3. Simplified31.1

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

  4. Final simplification15.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.320893942837389302245904465927389681568 \cdot 10^{-9} \lor \neg \left(\varepsilon \le 9.805431973736002190838386379022166561281 \cdot 10^{-10}\right):\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \left(\sin \varepsilon - 0\right) + \cos x\right)\right) + \mathsf{fma}\left(-\cos x, 1, \cos x \cdot 1\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019362 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))