Average Error: 39.9 → 0.7
Time: 21.4s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -7.379572630205786 \cdot 10^{-09}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 1.3301911003754664 \cdot 10^{-05}:\\ \;\;\;\;\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \left(-2 \cdot \sin \left(\mathsf{fma}\left(\varepsilon, \frac{1}{2}, x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \mathsf{fma}\left(\sin \varepsilon, \sin x, \cos x\right)\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -7.379572630205786 \cdot 10^{-09}:\\
\;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\

\mathbf{elif}\;\varepsilon \le 1.3301911003754664 \cdot 10^{-05}:\\
\;\;\;\;\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \left(-2 \cdot \sin \left(\mathsf{fma}\left(\varepsilon, \frac{1}{2}, x\right)\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 r984320 = x;
        double r984321 = eps;
        double r984322 = r984320 + r984321;
        double r984323 = cos(r984322);
        double r984324 = cos(r984320);
        double r984325 = r984323 - r984324;
        return r984325;
}
↓
double f(double x, double eps) {
        double r984326 = eps;
        double r984327 = -7.379572630205786e-09;
        bool r984328 = r984326 <= r984327;
        double r984329 = x;
        double r984330 = cos(r984329);
        double r984331 = cos(r984326);
        double r984332 = r984330 * r984331;
        double r984333 = sin(r984329);
        double r984334 = sin(r984326);
        double r984335 = r984333 * r984334;
        double r984336 = r984332 - r984335;
        double r984337 = r984336 - r984330;
        double r984338 = 1.3301911003754664e-05;
        bool r984339 = r984326 <= r984338;
        double r984340 = 0.5;
        double r984341 = r984340 * r984326;
        double r984342 = sin(r984341);
        double r984343 = -2.0;
        double r984344 = fma(r984326, r984340, r984329);
        double r984345 = sin(r984344);
        double r984346 = r984343 * r984345;
        double r984347 = r984342 * r984346;
        double r984348 = fma(r984334, r984333, r984330);
        double r984349 = r984332 - r984348;
        double r984350 = r984339 ? r984347 : r984349;
        double r984351 = r984328 ? r984337 : r984350;
        return r984351;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 3 regimes

  2. if eps < -7.379572630205786e-09

    1. Initial program 30.7

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

    3. Applied cos-sum1.3

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

    if -7.379572630205786e-09 < eps < 1.3301911003754664e-05

    1. Initial program 49.6

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

    3. Applied diff-cos37.8

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

    4. Simplified0.4

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

    5. Taylor expanded around -inf 0.4

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

    6. Simplified0.4

      \[\leadsto \color{blue}{\left(-2 \cdot \sin \left(\mathsf{fma}\left(\varepsilon, \frac{1}{2}, x\right)\right)\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)}\]

    if 1.3301911003754664e-05 < eps

    1. Initial program 30.6

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

    3. Applied cos-sum0.9

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

    4. Applied associate--l-1.0

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

    5. Simplified0.9

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -7.379572630205786 \cdot 10^{-09}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 1.3301911003754664 \cdot 10^{-05}:\\ \;\;\;\;\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \left(-2 \cdot \sin \left(\mathsf{fma}\left(\varepsilon, \frac{1}{2}, x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \mathsf{fma}\left(\sin \varepsilon, \sin x, \cos x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019135 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))