Average Error: 39.9 → 0.8
Time: 22.4s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -0.01398185726995361012614527140840436914004:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)\\ \mathbf{elif}\;\varepsilon \le 2.559402259473403249334308440232632619882 \cdot 10^{-5}:\\ \;\;\;\;\left(-2 \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -0.01398185726995361012614527140840436914004:\\
\;\;\;\;\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)\\

\mathbf{elif}\;\varepsilon \le 2.559402259473403249334308440232632619882 \cdot 10^{-5}:\\
\;\;\;\;\left(-2 \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\\

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

\end{array}
double f(double x, double eps) {
        double r30826 = x;
        double r30827 = eps;
        double r30828 = r30826 + r30827;
        double r30829 = cos(r30828);
        double r30830 = cos(r30826);
        double r30831 = r30829 - r30830;
        return r30831;
}
↓
double f(double x, double eps) {
        double r30832 = eps;
        double r30833 = -0.01398185726995361;
        bool r30834 = r30832 <= r30833;
        double r30835 = x;
        double r30836 = cos(r30835);
        double r30837 = cos(r30832);
        double r30838 = r30836 * r30837;
        double r30839 = sin(r30835);
        double r30840 = sin(r30832);
        double r30841 = r30839 * r30840;
        double r30842 = r30841 + r30836;
        double r30843 = r30838 - r30842;
        double r30844 = 2.5594022594734032e-05;
        bool r30845 = r30832 <= r30844;
        double r30846 = -2.0;
        double r30847 = 2.0;
        double r30848 = r30832 / r30847;
        double r30849 = sin(r30848);
        double r30850 = r30846 * r30849;
        double r30851 = r30835 + r30832;
        double r30852 = r30851 + r30835;
        double r30853 = r30852 / r30847;
        double r30854 = sin(r30853);
        double r30855 = r30850 * r30854;
        double r30856 = r30838 - r30841;
        double r30857 = r30856 - r30836;
        double r30858 = r30845 ? r30855 : r30857;
        double r30859 = r30834 ? r30843 : r30858;
        return r30859;
}

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 3 regimes

  2. if eps < -0.01398185726995361

    1. Initial program 30.0

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

    3. Applied cos-sum0.8

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

    4. Applied associate--l-0.8

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

    if -0.01398185726995361 < eps < 2.5594022594734032e-05

    1. Initial program 49.2

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

    3. Applied diff-cos38.1

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

    4. Simplified0.6

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

    6. Applied associate-*r*0.6

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

    if 2.5594022594734032e-05 < eps

    1. Initial program 31.5

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

    3. Applied cos-sum1.0

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -0.01398185726995361012614527140840436914004:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)\\ \mathbf{elif}\;\varepsilon \le 2.559402259473403249334308440232632619882 \cdot 10^{-5}:\\ \;\;\;\;\left(-2 \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]

Reproduce

herbie shell --seed 2019306 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))