Average Error: 39.5 → 0.7
Time: 16.7s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -0.22306452923435388:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \mathsf{fma}\left(\sin \varepsilon, \sin x, \cos x\right)\\ \mathbf{elif}\;\varepsilon \le 0.00391800888554098:\\ \;\;\;\;\sin \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right) \cdot \left(-2 \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \mathsf{fma}\left(\sin \varepsilon, \sin x, \cos x\right)\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -0.22306452923435388:\\
\;\;\;\;\cos x \cdot \cos \varepsilon - \mathsf{fma}\left(\sin \varepsilon, \sin x, \cos x\right)\\

\mathbf{elif}\;\varepsilon \le 0.00391800888554098:\\
\;\;\;\;\sin \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right) \cdot \left(-2 \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\\

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

\end{array}
double f(double x, double eps) {
        double r704247 = x;
        double r704248 = eps;
        double r704249 = r704247 + r704248;
        double r704250 = cos(r704249);
        double r704251 = cos(r704247);
        double r704252 = r704250 - r704251;
        return r704252;
}
↓
double f(double x, double eps) {
        double r704253 = eps;
        double r704254 = -0.22306452923435388;
        bool r704255 = r704253 <= r704254;
        double r704256 = x;
        double r704257 = cos(r704256);
        double r704258 = cos(r704253);
        double r704259 = r704257 * r704258;
        double r704260 = sin(r704253);
        double r704261 = sin(r704256);
        double r704262 = fma(r704260, r704261, r704257);
        double r704263 = r704259 - r704262;
        double r704264 = 0.00391800888554098;
        bool r704265 = r704253 <= r704264;
        double r704266 = 2.0;
        double r704267 = fma(r704266, r704256, r704253);
        double r704268 = r704267 / r704266;
        double r704269 = sin(r704268);
        double r704270 = -2.0;
        double r704271 = r704253 / r704266;
        double r704272 = sin(r704271);
        double r704273 = r704270 * r704272;
        double r704274 = r704269 * r704273;
        double r704275 = r704265 ? r704274 : r704263;
        double r704276 = r704255 ? r704263 : r704275;
        return r704276;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if eps < -0.22306452923435388 or 0.00391800888554098 < eps

    1. Initial program 29.7

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

    3. Applied cos-sum0.8

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

    4. Applied associate--l-0.8

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

    5. Simplified0.8

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

    if -0.22306452923435388 < eps < 0.00391800888554098

    1. Initial program 49.0

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

    3. Applied diff-cos37.2

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

    4. Simplified0.6

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

    6. Applied associate-*r*0.6

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -0.22306452923435388:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \mathsf{fma}\left(\sin \varepsilon, \sin x, \cos x\right)\\ \mathbf{elif}\;\varepsilon \le 0.00391800888554098:\\ \;\;\;\;\sin \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right) \cdot \left(-2 \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \mathsf{fma}\left(\sin \varepsilon, \sin x, \cos x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019156 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))