Average Error: 40.0 → 16.4
Time: 18.4s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.821618172153552 \cdot 10^{-16} \lor \neg \left(\varepsilon \le 4.8191518634182737 \cdot 10^{-8}\right):\\ \;\;\;\;\cos \varepsilon \cdot \cos x - \sqrt[3]{{\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{24} \cdot {\varepsilon}^{4} - \left(x \cdot \varepsilon + \frac{1}{2} \cdot {\varepsilon}^{2}\right)\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -1.821618172153552 \cdot 10^{-16} \lor \neg \left(\varepsilon \le 4.8191518634182737 \cdot 10^{-8}\right):\\
\;\;\;\;\cos \varepsilon \cdot \cos x - \sqrt[3]{{\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^{3}}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{24} \cdot {\varepsilon}^{4} - \left(x \cdot \varepsilon + \frac{1}{2} \cdot {\varepsilon}^{2}\right)\\

\end{array}
double f(double x, double eps) {
        double r255 = x;
        double r256 = eps;
        double r257 = r255 + r256;
        double r258 = cos(r257);
        double r259 = cos(r255);
        double r260 = r258 - r259;
        return r260;
}
↓
double f(double x, double eps) {
        double r261 = eps;
        double r262 = -1.8216181721535518e-16;
        bool r263 = r261 <= r262;
        double r264 = 4.819151863418274e-08;
        bool r265 = r261 <= r264;
        double r266 = !r265;
        bool r267 = r263 || r266;
        double r268 = cos(r261);
        double r269 = x;
        double r270 = cos(r269);
        double r271 = r268 * r270;
        double r272 = sin(r269);
        double r273 = sin(r261);
        double r274 = r272 * r273;
        double r275 = r274 + r270;
        double r276 = 3.0;
        double r277 = pow(r275, r276);
        double r278 = cbrt(r277);
        double r279 = r271 - r278;
        double r280 = 0.041666666666666664;
        double r281 = 4.0;
        double r282 = pow(r261, r281);
        double r283 = r280 * r282;
        double r284 = r269 * r261;
        double r285 = 0.5;
        double r286 = 2.0;
        double r287 = pow(r261, r286);
        double r288 = r285 * r287;
        double r289 = r284 + r288;
        double r290 = r283 - r289;
        double r291 = r267 ? r279 : r290;
        return r291;
}

Error

Bits error versus x

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if eps < -1.8216181721535518e-16 or 4.819151863418274e-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. Taylor expanded around inf 1.9

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

    6. Applied add-cbrt-cube2.1

      \[\leadsto \cos \varepsilon \cdot \cos x - \color{blue}{\sqrt[3]{\left(\left(\sin x \cdot \sin \varepsilon + \cos x\right) \cdot \left(\sin x \cdot \sin \varepsilon + \cos x\right)\right) \cdot \left(\sin x \cdot \sin \varepsilon + \cos x\right)}}\]

    7. Simplified2.0

      \[\leadsto \cos \varepsilon \cdot \cos x - \sqrt[3]{\color{blue}{{\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^{3}}}\]

    if -1.8216181721535518e-16 < eps < 4.819151863418274e-08

    1. Initial program 49.7

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

    3. Applied cos-sum49.6

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

    4. Taylor expanded around inf 49.6

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

    5. Taylor expanded around 0 32.2

      \[\leadsto \color{blue}{\frac{1}{24} \cdot {\varepsilon}^{4} - \left(x \cdot \varepsilon + \frac{1}{2} \cdot {\varepsilon}^{2}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification16.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.821618172153552 \cdot 10^{-16} \lor \neg \left(\varepsilon \le 4.8191518634182737 \cdot 10^{-8}\right):\\ \;\;\;\;\cos \varepsilon \cdot \cos x - \sqrt[3]{{\left(\sin x \cdot \sin \varepsilon + \cos x\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{24} \cdot {\varepsilon}^{4} - \left(x \cdot \varepsilon + \frac{1}{2} \cdot {\varepsilon}^{2}\right)\\ \end{array}\]

Reproduce

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