Average Error: 39.5 → 0.8
Time: 21.3s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.082410925969761437710064955197708513879 \cdot 10^{-4}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 1.289679011953039865754928189334194144067 \cdot 10^{-5}:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -1.082410925969761437710064955197708513879 \cdot 10^{-4}:\\
\;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\

\mathbf{elif}\;\varepsilon \le 1.289679011953039865754928189334194144067 \cdot 10^{-5}:\\
\;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\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 r1967750 = x;
        double r1967751 = eps;
        double r1967752 = r1967750 + r1967751;
        double r1967753 = cos(r1967752);
        double r1967754 = cos(r1967750);
        double r1967755 = r1967753 - r1967754;
        return r1967755;
}
↓
double f(double x, double eps) {
        double r1967756 = eps;
        double r1967757 = -0.00010824109259697614;
        bool r1967758 = r1967756 <= r1967757;
        double r1967759 = x;
        double r1967760 = cos(r1967759);
        double r1967761 = cos(r1967756);
        double r1967762 = r1967760 * r1967761;
        double r1967763 = sin(r1967759);
        double r1967764 = sin(r1967756);
        double r1967765 = r1967763 * r1967764;
        double r1967766 = r1967762 - r1967765;
        double r1967767 = r1967766 - r1967760;
        double r1967768 = 1.2896790119530399e-05;
        bool r1967769 = r1967756 <= r1967768;
        double r1967770 = -2.0;
        double r1967771 = 2.0;
        double r1967772 = r1967756 / r1967771;
        double r1967773 = sin(r1967772);
        double r1967774 = r1967759 + r1967756;
        double r1967775 = r1967774 + r1967759;
        double r1967776 = r1967775 / r1967771;
        double r1967777 = sin(r1967776);
        double r1967778 = r1967773 * r1967777;
        double r1967779 = r1967770 * r1967778;
        double r1967780 = r1967769 ? r1967779 : r1967767;
        double r1967781 = r1967758 ? r1967767 : r1967780;
        return r1967781;
}

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 < -0.00010824109259697614 or 1.2896790119530399e-05 < eps

    1. Initial program 30.2

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

    3. Applied cos-sum1.0

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

    if -0.00010824109259697614 < eps < 1.2896790119530399e-05

    1. Initial program 49.4

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

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.082410925969761437710064955197708513879 \cdot 10^{-4}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 1.289679011953039865754928189334194144067 \cdot 10^{-5}:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]

Reproduce

herbie shell --seed 2019169 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))