Average Error: 39.8 → 0.7
Time: 14.1s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -7.45592842694085654 \cdot 10^{-5} \lor \neg \left(\varepsilon \le 9.949689884521833 \cdot 10^{-6}\right):\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -7.45592842694085654 \cdot 10^{-5} \lor \neg \left(\varepsilon \le 9.949689884521833 \cdot 10^{-6}\right):\\
\;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\

\mathbf{else}:\\
\;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\

\end{array}
double f(double x, double eps) {
        double r48237 = x;
        double r48238 = eps;
        double r48239 = r48237 + r48238;
        double r48240 = cos(r48239);
        double r48241 = cos(r48237);
        double r48242 = r48240 - r48241;
        return r48242;
}
↓
double f(double x, double eps) {
        double r48243 = eps;
        double r48244 = -7.455928426940857e-05;
        bool r48245 = r48243 <= r48244;
        double r48246 = 9.949689884521833e-06;
        bool r48247 = r48243 <= r48246;
        double r48248 = !r48247;
        bool r48249 = r48245 || r48248;
        double r48250 = x;
        double r48251 = cos(r48250);
        double r48252 = cos(r48243);
        double r48253 = r48251 * r48252;
        double r48254 = sin(r48250);
        double r48255 = sin(r48243);
        double r48256 = r48254 * r48255;
        double r48257 = r48253 - r48256;
        double r48258 = r48257 - r48251;
        double r48259 = -2.0;
        double r48260 = 2.0;
        double r48261 = r48243 / r48260;
        double r48262 = sin(r48261);
        double r48263 = r48250 + r48243;
        double r48264 = r48263 + r48250;
        double r48265 = r48264 / r48260;
        double r48266 = sin(r48265);
        double r48267 = r48262 * r48266;
        double r48268 = r48259 * r48267;
        double r48269 = r48249 ? r48258 : r48268;
        return r48269;
}

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 < -7.455928426940857e-05 or 9.949689884521833e-06 < eps

    1. Initial program 30.3

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

    3. Applied cos-sum0.9

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

    if -7.455928426940857e-05 < eps < 9.949689884521833e-06

    1. Initial program 49.6

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

    3. Applied diff-cos38.0

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

    4. Simplified0.5

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -7.45592842694085654 \cdot 10^{-5} \lor \neg \left(\varepsilon \le 9.949689884521833 \cdot 10^{-6}\right):\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))