Average Error: 39.8 → 0.9
Time: 17.1s
Precision: 64
\[\cos \left(x + \varepsilon\right) - \cos x\]
↓
\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1170.229056579154985229251906275749206543:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \mathsf{fma}\left(\sin \varepsilon, \sin x, \cos x\right)\\ \mathbf{elif}\;\varepsilon \le 7051.556907543755187361966818571090698242:\\ \;\;\;\;\sin \left(\frac{\mathsf{fma}\left(x, 2, \varepsilon\right)}{2}\right) \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot -2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin \varepsilon \cdot \sin x\right) - \cos x\\ \end{array}\]
\cos \left(x + \varepsilon\right) - \cos x
↓
\begin{array}{l}
\mathbf{if}\;\varepsilon \le -1170.229056579154985229251906275749206543:\\
\;\;\;\;\cos x \cdot \cos \varepsilon - \mathsf{fma}\left(\sin \varepsilon, \sin x, \cos x\right)\\

\mathbf{elif}\;\varepsilon \le 7051.556907543755187361966818571090698242:\\
\;\;\;\;\sin \left(\frac{\mathsf{fma}\left(x, 2, \varepsilon\right)}{2}\right) \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot -2\right)\\

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

\end{array}
double f(double x, double eps) {
        double r22950 = x;
        double r22951 = eps;
        double r22952 = r22950 + r22951;
        double r22953 = cos(r22952);
        double r22954 = cos(r22950);
        double r22955 = r22953 - r22954;
        return r22955;
}
↓
double f(double x, double eps) {
        double r22956 = eps;
        double r22957 = -1170.229056579155;
        bool r22958 = r22956 <= r22957;
        double r22959 = x;
        double r22960 = cos(r22959);
        double r22961 = cos(r22956);
        double r22962 = r22960 * r22961;
        double r22963 = sin(r22956);
        double r22964 = sin(r22959);
        double r22965 = fma(r22963, r22964, r22960);
        double r22966 = r22962 - r22965;
        double r22967 = 7051.556907543755;
        bool r22968 = r22956 <= r22967;
        double r22969 = 2.0;
        double r22970 = fma(r22959, r22969, r22956);
        double r22971 = r22970 / r22969;
        double r22972 = sin(r22971);
        double r22973 = r22956 / r22969;
        double r22974 = sin(r22973);
        double r22975 = -2.0;
        double r22976 = r22974 * r22975;
        double r22977 = r22972 * r22976;
        double r22978 = r22963 * r22964;
        double r22979 = r22962 - r22978;
        double r22980 = r22979 - r22960;
        double r22981 = r22968 ? r22977 : r22980;
        double r22982 = r22958 ? r22966 : r22981;
        return r22982;
}

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 3 regimes

  2. if eps < -1170.229056579155

    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

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

    5. Simplified0.8

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

    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

      \[\leadsto -2 \cdot \color{blue}{\left(\sin \left(\frac{\varepsilon + 0}{2}\right) \cdot \sin \left(\frac{x + \left(x + \varepsilon\right)}{2}\right)\right)}\]
    5. Using strategy rm

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

      \[\leadsto \color{blue}{\left(1 \cdot -2\right)} \cdot \left(\sin \left(\frac{\varepsilon + 0}{2}\right) \cdot \sin \left(\frac{x + \left(x + \varepsilon\right)}{2}\right)\right)\]

    7. Applied associate-*l*1.1

      \[\leadsto \color{blue}{1 \cdot \left(-2 \cdot \left(\sin \left(\frac{\varepsilon + 0}{2}\right) \cdot \sin \left(\frac{x + \left(x + \varepsilon\right)}{2}\right)\right)\right)}\]

    8. Simplified1.1

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

    if 7051.556907543755 < eps

    1. Initial program 30.5

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

    3. Applied cos-sum0.8

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

    4. Simplified0.8

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

  4. Final simplification0.9

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

Reproduce

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