Average Error: 39.9 → 16.5
Time: 6.5s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -8.18910650708970111 \cdot 10^{-9} \lor \neg \left(\varepsilon \le 1.4188596036111877 \cdot 10^{-4}\right):\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \log \left(e^{\sin x \cdot \sin \varepsilon + \cos x}\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\ \end{array}\]
\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):\\
\;\;\;\;\cos x \cdot \cos \varepsilon - \log \left(e^{\sin x \cdot \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 r61310 = x;
        double r61311 = eps;
        double r61312 = r61310 + r61311;
        double r61313 = cos(r61312);
        double r61314 = cos(r61310);
        double r61315 = r61313 - r61314;
        return r61315;
}
↓
double f(double x, double eps) {
        double r61316 = eps;
        double r61317 = -8.189106507089701e-09;
        bool r61318 = r61316 <= r61317;
        double r61319 = 0.00014188596036111877;
        bool r61320 = r61316 <= r61319;
        double r61321 = !r61320;
        bool r61322 = r61318 || r61321;
        double r61323 = x;
        double r61324 = cos(r61323);
        double r61325 = cos(r61316);
        double r61326 = r61324 * r61325;
        double r61327 = sin(r61323);
        double r61328 = sin(r61316);
        double r61329 = r61327 * r61328;
        double r61330 = r61329 + r61324;
        double r61331 = exp(r61330);
        double r61332 = log(r61331);
        double r61333 = r61326 - r61332;
        double r61334 = 0.16666666666666666;
        double r61335 = 3.0;
        double r61336 = pow(r61323, r61335);
        double r61337 = r61334 * r61336;
        double r61338 = r61337 - r61323;
        double r61339 = 0.5;
        double r61340 = r61316 * r61339;
        double r61341 = r61338 - r61340;
        double r61342 = r61316 * r61341;
        double r61343 = r61322 ? r61333 : r61342;
        return r61343;
}

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 < -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. Using strategy rm

    6. Applied add-log-exp1.2

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

    7. Applied add-log-exp1.2

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

    8. Applied sum-log1.3

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

    9. Simplified1.2

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

    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

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

Reproduce

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