Average Error: 39.1 → 0.7
Time: 24.6s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -2.68449879533306243542534197388249594951 \cdot 10^{-4} \lor \neg \left(\varepsilon \le 3.028675491606891100851602083299241030545 \cdot 10^{-6}\right):\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -2.68449879533306243542534197388249594951 \cdot 10^{-4} \lor \neg \left(\varepsilon \le 3.028675491606891100851602083299241030545 \cdot 10^{-6}\right):\\
\;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\

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

\end{array}
double f(double x, double eps) {
        double r35016 = x;
        double r35017 = eps;
        double r35018 = r35016 + r35017;
        double r35019 = cos(r35018);
        double r35020 = cos(r35016);
        double r35021 = r35019 - r35020;
        return r35021;
}
↓
double f(double x, double eps) {
        double r35022 = eps;
        double r35023 = -0.00026844987953330624;
        bool r35024 = r35022 <= r35023;
        double r35025 = 3.028675491606891e-06;
        bool r35026 = r35022 <= r35025;
        double r35027 = !r35026;
        bool r35028 = r35024 || r35027;
        double r35029 = x;
        double r35030 = cos(r35029);
        double r35031 = cos(r35022);
        double r35032 = r35030 * r35031;
        double r35033 = sin(r35029);
        double r35034 = sin(r35022);
        double r35035 = r35033 * r35034;
        double r35036 = r35032 - r35035;
        double r35037 = r35036 - r35030;
        double r35038 = -2.0;
        double r35039 = 2.0;
        double r35040 = r35022 / r35039;
        double r35041 = sin(r35040);
        double r35042 = r35029 + r35022;
        double r35043 = r35042 + r35029;
        double r35044 = r35043 / r35039;
        double r35045 = sin(r35044);
        double r35046 = r35041 * r35045;
        double r35047 = r35038 * r35046;
        double r35048 = r35028 ? r35037 : r35047;
        return r35048;
}

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 < -0.00026844987953330624 or 3.028675491606891e-06 < eps

    1. Initial program 29.4

      \[\cos \left(x + \varepsilon\right) - \cos x\]
    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 -0.00026844987953330624 < eps < 3.028675491606891e-06

    1. Initial program 49.2

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

    3. Applied diff-cos38.3

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

    4. Simplified0.5

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -2.68449879533306243542534197388249594951 \cdot 10^{-4} \lor \neg \left(\varepsilon \le 3.028675491606891100851602083299241030545 \cdot 10^{-6}\right):\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))