Average Error: 39.9 → 16.5
Time: 6.9s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -8.18910650708970111 \cdot 10^{-9} \lor \neg \left(\varepsilon \le 1.4188596036111877 \cdot 10^{-4}\right):\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\cos x \cdot \cos \varepsilon\right)\right) - \mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -8.18910650708970111 \cdot 10^{-9} \lor \neg \left(\varepsilon \le 1.4188596036111877 \cdot 10^{-4}\right):\\
\;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\cos x \cdot \cos \varepsilon\right)\right) - \mathsf{fma}\left(\sin x, \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 r51295 = x;
        double r51296 = eps;
        double r51297 = r51295 + r51296;
        double r51298 = cos(r51297);
        double r51299 = cos(r51295);
        double r51300 = r51298 - r51299;
        return r51300;
}
↓
double f(double x, double eps) {
        double r51301 = eps;
        double r51302 = -8.189106507089701e-09;
        bool r51303 = r51301 <= r51302;
        double r51304 = 0.00014188596036111877;
        bool r51305 = r51301 <= r51304;
        double r51306 = !r51305;
        bool r51307 = r51303 || r51306;
        double r51308 = x;
        double r51309 = cos(r51308);
        double r51310 = cos(r51301);
        double r51311 = r51309 * r51310;
        double r51312 = expm1(r51311);
        double r51313 = log1p(r51312);
        double r51314 = sin(r51308);
        double r51315 = sin(r51301);
        double r51316 = fma(r51314, r51315, r51309);
        double r51317 = r51313 - r51316;
        double r51318 = 0.16666666666666666;
        double r51319 = 3.0;
        double r51320 = pow(r51308, r51319);
        double r51321 = r51318 * r51320;
        double r51322 = r51321 - r51308;
        double r51323 = 0.5;
        double r51324 = r51301 * r51323;
        double r51325 = r51322 - r51324;
        double r51326 = r51301 * r51325;
        double r51327 = r51307 ? r51317 : r51326;
        return r51327;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if eps < -8.189106507089701e-09 or 0.00014188596036111877 < eps

    1. Initial program 30.9

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

    3. Applied cos-sum1.1

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

    4. Applied associate--l-1.1

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

    5. Simplified1.1

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

    7. Applied log1p-expm1-u1.2

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos x \cdot \cos \varepsilon\right)\right)} - \mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\]

    if -8.189106507089701e-09 < eps < 0.00014188596036111877

    1. Initial program 49.3

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

    2. Taylor expanded around 0 32.4

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

    3. Simplified32.4

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

  4. Final simplification16.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -8.18910650708970111 \cdot 10^{-9} \lor \neg \left(\varepsilon \le 1.4188596036111877 \cdot 10^{-4}\right):\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\cos x \cdot \cos \varepsilon\right)\right) - \mathsf{fma}\left(\sin x, \sin \varepsilon, \cos x\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020049 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))