Average Error: 39.6 → 1.3
Time: 22.6s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -5770898450817316028416:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 0.01574298965693270158094918542701634578407:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -5770898450817316028416:\\
\;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\

\mathbf{elif}\;\varepsilon \le 0.01574298965693270158094918542701634578407:\\
\;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\

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

\end{array}
double f(double x, double eps) {
        double r22809 = x;
        double r22810 = eps;
        double r22811 = r22809 + r22810;
        double r22812 = cos(r22811);
        double r22813 = cos(r22809);
        double r22814 = r22812 - r22813;
        return r22814;
}
↓
double f(double x, double eps) {
        double r22815 = eps;
        double r22816 = -5.770898450817316e+21;
        bool r22817 = r22815 <= r22816;
        double r22818 = x;
        double r22819 = cos(r22818);
        double r22820 = cos(r22815);
        double r22821 = r22819 * r22820;
        double r22822 = sin(r22818);
        double r22823 = sin(r22815);
        double r22824 = r22822 * r22823;
        double r22825 = r22821 - r22824;
        double r22826 = r22825 - r22819;
        double r22827 = 0.0157429896569327;
        bool r22828 = r22815 <= r22827;
        double r22829 = -2.0;
        double r22830 = 2.0;
        double r22831 = r22815 / r22830;
        double r22832 = sin(r22831);
        double r22833 = r22818 + r22815;
        double r22834 = r22833 + r22818;
        double r22835 = r22834 / r22830;
        double r22836 = sin(r22835);
        double r22837 = r22832 * r22836;
        double r22838 = r22829 * r22837;
        double r22839 = r22824 + r22819;
        double r22840 = r22821 - r22839;
        double r22841 = r22828 ? r22838 : r22840;
        double r22842 = r22817 ? r22826 : r22841;
        return r22842;
}

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 < -5.770898450817316e+21

    1. Initial program 30.5

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

    3. Applied cos-sum0.8

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

    if -5.770898450817316e+21 < eps < 0.0157429896569327

    1. Initial program 48.2

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

    3. Applied diff-cos37.2

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

    4. Simplified1.8

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

    if 0.0157429896569327 < eps

    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

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

  4. Final simplification1.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -5770898450817316028416:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 0.01574298965693270158094918542701634578407:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))