Average Error: 39.6 → 16.0
Time: 6.6s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.3976165695423843 \cdot 10^{-14}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)\\ \mathbf{elif}\;\varepsilon \le 2.63597490420144011 \cdot 10^{-8}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\left(\cos x \cdot \cos \varepsilon - \sin \varepsilon \cdot \sin x\right) - \cos x}\right)\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -1.3976165695423843 \cdot 10^{-14}:\\
\;\;\;\;\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)\\

\mathbf{elif}\;\varepsilon \le 2.63597490420144011 \cdot 10^{-8}:\\
\;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\

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

\end{array}
double f(double x, double eps) {
        double r52657 = x;
        double r52658 = eps;
        double r52659 = r52657 + r52658;
        double r52660 = cos(r52659);
        double r52661 = cos(r52657);
        double r52662 = r52660 - r52661;
        return r52662;
}
↓
double f(double x, double eps) {
        double r52663 = eps;
        double r52664 = -1.3976165695423843e-14;
        bool r52665 = r52663 <= r52664;
        double r52666 = x;
        double r52667 = cos(r52666);
        double r52668 = cos(r52663);
        double r52669 = r52667 * r52668;
        double r52670 = sin(r52666);
        double r52671 = sin(r52663);
        double r52672 = r52670 * r52671;
        double r52673 = r52672 + r52667;
        double r52674 = r52669 - r52673;
        double r52675 = 2.63597490420144e-08;
        bool r52676 = r52663 <= r52675;
        double r52677 = 0.16666666666666666;
        double r52678 = 3.0;
        double r52679 = pow(r52666, r52678);
        double r52680 = r52677 * r52679;
        double r52681 = r52680 - r52666;
        double r52682 = 0.5;
        double r52683 = r52663 * r52682;
        double r52684 = r52681 - r52683;
        double r52685 = r52663 * r52684;
        double r52686 = r52671 * r52670;
        double r52687 = r52669 - r52686;
        double r52688 = r52687 - r52667;
        double r52689 = exp(r52688);
        double r52690 = log(r52689);
        double r52691 = r52676 ? r52685 : r52690;
        double r52692 = r52665 ? r52674 : r52691;
        return r52692;
}

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 < -1.3976165695423843e-14

    1. Initial program 31.1

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

    3. Applied cos-sum2.1

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

    4. Applied associate--l-2.1

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

    if -1.3976165695423843e-14 < eps < 2.63597490420144e-08

    1. Initial program 49.3

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

    2. Taylor expanded around 0 31.5

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

    3. Simplified31.5

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

    if 2.63597490420144e-08 < eps

    1. Initial program 30.3

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

    5. Applied add-log-exp1.2

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

    6. Applied add-log-exp1.3

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

    7. Applied add-log-exp1.5

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

    8. Applied diff-log1.5

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

    9. Applied diff-log1.5

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

    10. Simplified1.3

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

    12. Applied *-commutative1.3

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

  4. Final simplification16.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.3976165695423843 \cdot 10^{-14}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)\\ \mathbf{elif}\;\varepsilon \le 2.63597490420144011 \cdot 10^{-8}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\frac{1}{6} \cdot {x}^{3} - x\right) - \varepsilon \cdot \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\left(\cos x \cdot \cos \varepsilon - \sin \varepsilon \cdot \sin x\right) - \cos x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020024 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))