Average Error: 39.7 → 16.1
Time: 6.9s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -3.658449891821199522487853550423153592419 \cdot 10^{-6} \lor \neg \left(\varepsilon \le 3.130998694967744134270873183900629754817 \cdot 10^{-8}\right):\\ \;\;\;\;\cos x \cdot \cos \varepsilon - 1 \cdot \left(\sin x \cdot \sin \varepsilon + \cos x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{24} \cdot {\varepsilon}^{4} - \left(x \cdot \varepsilon + \frac{1}{2} \cdot {\varepsilon}^{2}\right)\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -3.658449891821199522487853550423153592419 \cdot 10^{-6} \lor \neg \left(\varepsilon \le 3.130998694967744134270873183900629754817 \cdot 10^{-8}\right):\\
\;\;\;\;\cos x \cdot \cos \varepsilon - 1 \cdot \left(\sin x \cdot \sin \varepsilon + \cos x\right)\\

\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 r75442 = x;
        double r75443 = eps;
        double r75444 = r75442 + r75443;
        double r75445 = cos(r75444);
        double r75446 = cos(r75442);
        double r75447 = r75445 - r75446;
        return r75447;
}
↓
double f(double x, double eps) {
        double r75448 = eps;
        double r75449 = -3.6584498918211995e-06;
        bool r75450 = r75448 <= r75449;
        double r75451 = 3.130998694967744e-08;
        bool r75452 = r75448 <= r75451;
        double r75453 = !r75452;
        bool r75454 = r75450 || r75453;
        double r75455 = x;
        double r75456 = cos(r75455);
        double r75457 = cos(r75448);
        double r75458 = r75456 * r75457;
        double r75459 = 1.0;
        double r75460 = sin(r75455);
        double r75461 = sin(r75448);
        double r75462 = r75460 * r75461;
        double r75463 = r75462 + r75456;
        double r75464 = r75459 * r75463;
        double r75465 = r75458 - r75464;
        double r75466 = 0.041666666666666664;
        double r75467 = 4.0;
        double r75468 = pow(r75448, r75467);
        double r75469 = r75466 * r75468;
        double r75470 = r75455 * r75448;
        double r75471 = 0.5;
        double r75472 = 2.0;
        double r75473 = pow(r75448, r75472);
        double r75474 = r75471 * r75473;
        double r75475 = r75470 + r75474;
        double r75476 = r75469 - r75475;
        double r75477 = r75454 ? r75465 : r75476;
        return r75477;
}

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 < -3.6584498918211995e-06 or 3.130998694967744e-08 < eps

    1. Initial program 30.1

      \[\cos \left(x + \varepsilon\right) - \cos x\]
    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 *-un-lft-identity1.1

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

    if -3.6584498918211995e-06 < eps < 3.130998694967744e-08

    1. Initial program 49.7

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

    3. Applied cos-sum49.3

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

    4. Applied associate--l-49.3

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

    6. Applied *-un-lft-identity49.3

      \[\leadsto \cos x \cdot \cos \varepsilon - \color{blue}{1 \cdot \left(\sin x \cdot \sin \varepsilon + \cos x\right)}\]
    7. Using strategy rm

    8. Applied add-log-exp49.5

      \[\leadsto \cos x \cdot \cos \varepsilon - \color{blue}{\log \left(e^{1 \cdot \left(\sin x \cdot \sin \varepsilon + \cos x\right)}\right)}\]

    9. Applied add-log-exp49.3

      \[\leadsto \color{blue}{\log \left(e^{\cos x \cdot \cos \varepsilon}\right)} - \log \left(e^{1 \cdot \left(\sin x \cdot \sin \varepsilon + \cos x\right)}\right)\]

    10. Applied diff-log49.3

      \[\leadsto \color{blue}{\log \left(\frac{e^{\cos x \cdot \cos \varepsilon}}{e^{1 \cdot \left(\sin x \cdot \sin \varepsilon + \cos x\right)}}\right)}\]

    11. Simplified49.3

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

    12. Taylor expanded around 0 31.8

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

  4. Final simplification16.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -3.658449891821199522487853550423153592419 \cdot 10^{-6} \lor \neg \left(\varepsilon \le 3.130998694967744134270873183900629754817 \cdot 10^{-8}\right):\\ \;\;\;\;\cos x \cdot \cos \varepsilon - 1 \cdot \left(\sin x \cdot \sin \varepsilon + \cos x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{24} \cdot {\varepsilon}^{4} - \left(x \cdot \varepsilon + \frac{1}{2} \cdot {\varepsilon}^{2}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019347 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))