Average Error: 39.6 → 16.4
Time: 7.5s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -2.60552829839509202 \cdot 10^{-14} \lor \neg \left(\varepsilon \le 2.832408619440934 \cdot 10^{-8}\right):\\ \;\;\;\;\log \left(e^{\cos x \cdot \cos \varepsilon}\right) + \left(\left(-\sin x \cdot \sin \varepsilon\right) - \cos x\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -2.60552829839509202 \cdot 10^{-14} \lor \neg \left(\varepsilon \le 2.832408619440934 \cdot 10^{-8}\right):\\
\;\;\;\;\log \left(e^{\cos x \cdot \cos \varepsilon}\right) + \left(\left(-\sin x \cdot \sin \varepsilon\right) - \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 r63770 = x;
        double r63771 = eps;
        double r63772 = r63770 + r63771;
        double r63773 = cos(r63772);
        double r63774 = cos(r63770);
        double r63775 = r63773 - r63774;
        return r63775;
}
↓
double f(double x, double eps) {
        double r63776 = eps;
        double r63777 = -2.605528298395092e-14;
        bool r63778 = r63776 <= r63777;
        double r63779 = 2.8324086194409338e-08;
        bool r63780 = r63776 <= r63779;
        double r63781 = !r63780;
        bool r63782 = r63778 || r63781;
        double r63783 = x;
        double r63784 = cos(r63783);
        double r63785 = cos(r63776);
        double r63786 = r63784 * r63785;
        double r63787 = exp(r63786);
        double r63788 = log(r63787);
        double r63789 = sin(r63783);
        double r63790 = sin(r63776);
        double r63791 = r63789 * r63790;
        double r63792 = -r63791;
        double r63793 = r63792 - r63784;
        double r63794 = r63788 + r63793;
        double r63795 = 0.16666666666666666;
        double r63796 = 3.0;
        double r63797 = pow(r63783, r63796);
        double r63798 = r63795 * r63797;
        double r63799 = r63798 - r63783;
        double r63800 = 0.5;
        double r63801 = r63776 * r63800;
        double r63802 = r63799 - r63801;
        double r63803 = r63776 * r63802;
        double r63804 = r63782 ? r63794 : r63803;
        return r63804;
}

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 < -2.605528298395092e-14 or 2.8324086194409338e-08 < eps

    1. Initial program 30.9

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

    3. Applied cos-sum1.6

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

    5. Applied sub-neg1.6

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

    6. Applied associate--l+1.6

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

    8. Applied add-log-exp1.8

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

    if -2.605528298395092e-14 < eps < 2.8324086194409338e-08

    1. Initial program 48.7

      \[\cos \left(x + \varepsilon\right) - \cos x\]

    2. Taylor expanded around 0 31.9

      \[\leadsto \color{blue}{\frac{1}{6} \cdot \left({x}^{3} \cdot \varepsilon\right) - \left(x \cdot \varepsilon + \frac{1}{2} \cdot {\varepsilon}^{2}\right)}\]

    3. Simplified31.9

      \[\leadsto \color{blue}{\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification16.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -2.60552829839509202 \cdot 10^{-14} \lor \neg \left(\varepsilon \le 2.832408619440934 \cdot 10^{-8}\right):\\ \;\;\;\;\log \left(e^{\cos x \cdot \cos \varepsilon}\right) + \left(\left(-\sin x \cdot \sin \varepsilon\right) - \cos x\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))