Average Error: 39.8 → 0.9
Time: 16.1s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1170.229056579154985229251906275749206543 \lor \neg \left(\varepsilon \le 7051.556907543755187361966818571090698242\right):\\ \;\;\;\;\left(\cos \varepsilon \cdot \cos x - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot -2\right) \cdot \sin \left(\frac{2 \cdot x + \varepsilon}{2}\right)\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -1170.229056579154985229251906275749206543 \lor \neg \left(\varepsilon \le 7051.556907543755187361966818571090698242\right):\\
\;\;\;\;\left(\cos \varepsilon \cdot \cos x - \sin x \cdot \sin \varepsilon\right) - \cos x\\

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

\end{array}
double f(double x, double eps) {
        double r22969 = x;
        double r22970 = eps;
        double r22971 = r22969 + r22970;
        double r22972 = cos(r22971);
        double r22973 = cos(r22969);
        double r22974 = r22972 - r22973;
        return r22974;
}
↓
double f(double x, double eps) {
        double r22975 = eps;
        double r22976 = -1170.229056579155;
        bool r22977 = r22975 <= r22976;
        double r22978 = 7051.556907543755;
        bool r22979 = r22975 <= r22978;
        double r22980 = !r22979;
        bool r22981 = r22977 || r22980;
        double r22982 = cos(r22975);
        double r22983 = x;
        double r22984 = cos(r22983);
        double r22985 = r22982 * r22984;
        double r22986 = sin(r22983);
        double r22987 = sin(r22975);
        double r22988 = r22986 * r22987;
        double r22989 = r22985 - r22988;
        double r22990 = r22989 - r22984;
        double r22991 = 2.0;
        double r22992 = r22975 / r22991;
        double r22993 = sin(r22992);
        double r22994 = -2.0;
        double r22995 = r22993 * r22994;
        double r22996 = r22991 * r22983;
        double r22997 = r22996 + r22975;
        double r22998 = r22997 / r22991;
        double r22999 = sin(r22998);
        double r23000 = r22995 * r22999;
        double r23001 = r22981 ? r22990 : r23000;
        return r23001;
}

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 < -1170.229056579155 or 7051.556907543755 < eps

    1. Initial program 30.2

      \[\cos \left(x + \varepsilon\right) - \cos x\]
    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. Simplified0.8

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

    if -1170.229056579155 < eps < 7051.556907543755

    1. Initial program 49.0

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

    3. Applied diff-cos37.7

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

    4. Simplified1.1

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

    6. Applied *-un-lft-identity1.1

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

    7. Applied associate-*l*1.1

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

    8. Simplified1.1

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1170.229056579154985229251906275749206543 \lor \neg \left(\varepsilon \le 7051.556907543755187361966818571090698242\right):\\ \;\;\;\;\left(\cos \varepsilon \cdot \cos x - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot -2\right) \cdot \sin \left(\frac{2 \cdot x + \varepsilon}{2}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019196 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))